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UNlVKKSn  Y  <^i  CAUFORNU 


ELEMEi^TS 


THEOET  OF  THE  NEWTONIAN  POTENTIAL 
.    FUNCTION. 


B.  O.  PEIRCE,  Ph.D., 

ASSISTANT   PROFESSOR   OF   MATHEMATICS   AND   PHTSIC8 
IX   HABVAKD   rXIYEBSITY. 


-ppX««>*-.- 


BOSTON: 

PUBLISHED  BY  GIXX  &  COMPAXY. 

1886. 


1054G0 


Entered,  according  to  Act  of  Congress,  in  the  year  1886,  by 

GINN   &  COMPAJSTY, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


J.  S.  CusHiNo  &  Co.,  Pbintkrs,  Bostoh. 


PKEFACE. 


This   book    is    almost    entirely    made    up   of    lecture-notes 

which    from    time    to    tiroe    during    the    last    four    ^-ears    I 

have  written   out   for   the    use  of    students  who   have   begun 

\       with  me   the   study  of   what   I   have   ventui-ed   to   call,    after 

^      Neumann,  the  Newtonian  Potential  Function. 

V  The  notes  were  intended  for  readers  somewhat  familiar  with 

the   principles   of  the  Differential  and  Integral  Calculus,    but 

unacquainted  with    many  (^  the   methods  commonly  used    in 

.  y     ^PP^'^^o    Mathematics    to    the    study   of    physical    problems. 

**^      These  students,   I  learned,  found  it  difficult  to  get  from  any 

single   book  in  English  a  treatment   of   the   subject   at   once 

elementary   enough   to    be   within    their   easy   comprehension, 

\i^      and  at  the  same  time  suit<Bd  to  the  purposes  of  such  of  them 

^     as    intended    eventually    to    pursue    the    subject    farther,  or 

wished,    without    necessarily   making   a  ,  '/V^?vtlty   of    Mathe- 

V      matical    Physics,    to    prepare    themselves    to    study    Experi- 

pV      mental   Physics    thoroughly    and    understandingly,       "What   is 

here    printed   seems   to   have    been  of   use    to  some  of   those 

who  have  read  it  in  manuscript,  and  it  is  hoped  that  it  may 

now  be  helpful  to  a  larger  number  of  students. 

Since  these  notes  are  professedly  elementary  in  character, 
I  feel  that  no  apology  is  needed  for  what  may  seem  to  be 
the  rather  prolix  way  in  which  some  of  the  subjects  are 
treated,  or   for   au    arrangement   of    matter   which   would   be 


IV  •  PREFACE. 

unsuitable  in  a  book  intended  for  a  different  class  of  readers. 
I  have  not  hesitated  to  use  a  long  proof  whenever  this  has 
seemed  to  me  more  easily  comprehensible  than  a  short  and 
mathematically  neater  one,  and  I  have  often  given  more  than 
one  demonstration  of  a  single  theorem  in  order  to  illustrate 
different  methods  of  work.  Although  I  have  used  freeh* 
the  notation  *  of  the  Calculus,  I  have  assumed  on  the  part 
of  the  reader  only  an  elementary  knowledge  of  its  principles. 

The  short  treatment  of  Electrostatics  in  Chapter  v.  is  in- 
troduced to  show  how  the  theorems  of  the  preceding  chapters 
may  be  used  in  solving  phj-sical  problems  ;  but  it  is  hoped  that 
a  person  who  has  mastered  even  the  little  here  given  will  be 
able  to  understand,  with  the  aid  of  some  good  treatise  on 
Experimental  Ph^'sics,  most  of  the  phenomena  of  Electro- 
statics.  It  is  also  hoped  that  those  readers  who  mean  to  study 
the   subject   of    Electricity   from    the    mathematical   point   of 

*  In  this  book  the  change  made  in  a  function  u  by  giving  to  the 
independent  variable  x  the  arbitrary,  finite  increment  Ax,  and  keeping  the 
other  independent  variables,  if  there  are  any,  unchanged,  is  denoted  by 
Ax".  Similarly,  Ayti  and  a^m  express  the  increments  of  u  due  to  changes 
respectively  in  1/  alone  and  in  z  alone.  The  total  change  in  u  due  to 
simultaneous  changes  in  all  the  independent  variables  is  sometimes 
denoted  by  Au;  so  that  if  u=/{x,  y,  z), 

A,u    .     ,  A„H     ^     ,  AzM     ^     , 

Au  =  — ^^  •  Ax  H !^  •  Ay  -\ Az-\-  e. 

Ax  Ay  Az 

where  «  is  an  infinitesimal  of  an  order  higher  than  the  first. 

The  partial  derivatives  of  u  with  respect  to  x,  y,  and  -  are  denoted  by 
DxW,  Dgii,  and  D^it,  and  the  sign  =  placed  between  a  variable  and  a  con- 
stant is  used  to  show  that  the  former  is  to  be  made  to  approach  the 
latter  as  its  limit.  In  those  cases  where  it  is  desirable  to  draw  attention 
to  the  fact  that  a  certain  derivative  is   total,  the   differential  notation 

—  IS  used. 
ax 


PREFACE.  V 

view  will  find  what  they  have  learned  here  useful  when 
they  take  up  standard  works  on  the  subject. 

My  sincere  thanks  are  due  to  H.  N.  Wheeler,  A.M.,  who 
has  read  much  of  the  manuscript  of  the  following  pages  and 
all  of  the  proof,  and  to  Dr.  E.  II.  Hall,  who  has  examined 
parts  of  Chapters  iv.  and  v.  and  helped  me  with  various 
suggestions.  I  am  indebted  to  other  friends  also,  and  among 
them  to  Mr.  "VY.  A.  Stone  for  the  use  of  some  of  his  problems. 

The  reader  who  wishes  to  get  a  thorough  knowledge  of 
the  properties  of  the  Newtonian  Potential  Function  and  of 
its  applications  to  problems  in  Electricity  is  referred  to  the 
following  Avorks,  which,  with  others,  I  have  consulted  and  used 
in  writing  these  notes. 

Betti :    Teorica    delle    Forze    Newtoniane   e    sue    Applicazioni    all' 

Elettrostatica  e  al  Magnetismo. 
Clausius  :  Die  Potentialf  unction  und  das  Potential. 
Cumming :  An  Introduction  to  the  Theory  of  Electricity. 
Chrystal:  Tlie  article    "Electricity"   in    the  Ninth    Edition  of   the 

Encyclopaedia  Britannica. 
Dirichlet:    Vorlesungen  liber   die  ini  umgekehrten  Yerhiiltniss  des 

Quadrats  der  Entfernung  wirkendeu  Kriifto. 
Gauss :    Allgemeine  Lehrsiitze  in  Beziehung  auf  die  ini  verkehrten 

Yerhaltnisse  des  Quadrates  der  Entfernung  Avirkenden  Anzieh- 

ungs-  und  Abstossungskriifte.      Also  other  papers  to  be  found 

in  Yolume  Y.  of  his  Gesammelte  Werke. 
Green  :   An  Essay  on  the  Application  of  Mathematical  Analysis  to 

the  Theories  of  Electricity  and  Magnetism.* 
Mascart:  Traite  d'Electricite  Statique.     Also  Wallentin's  translation 

of  the  same  work  into  German,  with  additions. 

*A  copy  of  the  original  edition  of  this  paper  is  to  be  found  in  tlie 
Library  of  Harvard  University,  Gore  Hall,  Cambridge.  Tiie  paper  has 
boon  reprinted  by  Ferrers  in  "'I'lie  Mathematical  Papers  of  George  (ireen," 
and  by  Thomson  in  Crelle's  Journal. 


vi  PREFACE. 

Mascart  et  Joubert:.  Lemons   sur  I'Electricite   et   le  Magnetisme. 

Also  Atkinson's  translation  of  the  same  work  into  English,  with 

additions. 
Mathieu:    Theorie   du   Potential    et    ses    Applications    h  I'Electro- 

statique  et  au  Magnetisme. 
Maxvrell:  An  Elementary  Treatise  on  Electricity.     A  Treatise  on 

Electricity  and  Magnetism. 
Minchin :  A  Treatise  on  Statics. 
C.  Neumann :  Untersuchungen  iiber  das  Logarithmische  und  New- 

ton'sche  Potential. 
Riemann:  Schwere,  Electricitat  und  Magnetismus,  edited  by  Hatten- 

dorff. 
Schell :  Theorie  der  Bewegung  und  der  Krafte. 
Thomson :  Keprint  of  Papers  on  Electrostatics  and  Magnetism. 
Thomson  and  Tait :  A  Treatise  on  Natural  Philosophy. 
Todhunter :  A  Treatise  on  Analytical  Statics. 
Watson  and  Burbury:    The  Mathematical  Theory  of   Electricity 

and  Magnetism. 
"Wiedemann :  Die  Lehre  von  der  Electricitat. 


TABLE    OF   CONTENTS. 


CHAPTER  I. 

THE  ATTRACTION  OP  GRAVITATION. 
Section.  Page 

1.  The  law  of  gravitation 1 

2.  The  attraction  at  a  point 1 

3.  The  unit  of  force 2 

4.  The  attraction  due  to  discrete  particles         ....  2 

5.  The  attraction  of  a  straight  wire  at  a  point  in  its  axis  .  3 

6.  The  attraction  at  any  point  due  to  a  straight  wire        .         .  4 

7.  The  attraction  at  a  point  in  its  axis  due  to  a  cylinder  of 

revolution       .........       7 

8.  The  attraction  at  the  vertex  of  a  cone  of   revolution   due 

to  the  whole  cone  and  to  different  frusta         ...       8 

9.  The  attraction  due  to  a  homogeneous   spherical  shell;  to 

a  solid  sphere  ........     11 

10.  The  attraction  due  to  a  homogeneous  hemisphere         .         .     13 

11.  Apparent   anomalies   in   the   latitudes   of    places   near  the 

foot  of  a  hemispherical  hill 15 

12.  The  attraction  due  to  any  ellipsoidal  homoeoid  is  zero  at 

all  points  within  the  cavity  enclosed  by  the  shell   .         .     16 

13.  The  attraction  due  to  a  spherical  shell  whose   density  at 

any  point  depends  upon  the  distance  of  the  point  from 
the  centre 18 

14.  The  attraction  at  any  point  due  to  any  given  mass       .         .     19 

15.  The  component  in  any  direction  of  the  attraction  at  a  point 

P  due  to  a  given  mass  is  always  finite    .         .         .         .21 


Viii  TABLE   OF   CONTENTS. 

Sectiok  Page 

16.  The  attraction  between  two  straight  wires    .         .         .         .22 

17.  The  attraction  between  two  spheres 23 

18.  The  attraction  between  any  two  rigid  bodies        .        .        .24 

CHAPTER  II. 

THE   NEWTONIAN   POTENTIAL   FUNCTION   IN   THE   CASE   OP 
GRAVITATION. 

19.  Definition  of  the  potential  function 29 

20.  The  derivatives  of   the  potential  function  relative  to  the 

space  coordinates  are  functions  of  these  coordinates 
which  represent  the  components  parallel  to  the  coordi- 
nate axes  of  the  attraction  at  the  point  (x,  y,  z)     .         .     30 

21.  Extension  of  the  statement  of  the  last  section       .         .         .31 

22.  The  potential  function  due  to  a  given  attracting  mass  is 

everywhere  finite,  and  the  statements  of  the  two  pre- 
ceding sections  hold  good  for  points  within  the  attract- 
ing mass 32 

23.  The  potential  function  due  to  a  straight  wire        .         .         .34 

24.  The  potential  function  due  to  a  spherical  shell      .         .         .35 

25.  Equipotential  surfaces'and  their  properties  .         .         .         .37 

26.  The  potential  function  is  zero  at  infinity       .         .         .         .40 

27.  The  potential  function  as  a  measure  of  work         .         .         .40 

28.  Laplace's  Equation 42 

29.  The  second  derivatives  of  the  potential  function  are  finite 

at  points  within  the  attracting  mass        .         .         .         .43 

30.  The  first  derivatives  of  the  potential  function  change  con- 

tinuously as  the  point,  (x,  y,  z)  moves  through  the 
boundaries  of  an  attracting  mass 48 

31.  Theorem  due  to  Gauss.     The  potential  function  can  have 

no  maxima  or  minima  at  points  of  empty  space     .        .  50 

32.  Tubes  of  force  and  their  properties 53 

33.  Spherical  distributions  of  matter  and  their  attractions  .  54 

34.  Cylindrical  distributions  of  matter  and  their  attractions      .  58 


TABLE   OF   CONTENTS.  IX 

Sbctiok  Page 

35.  Poisson's  Equation  obtained  by  the  application  of  Gauss's 

Theorem  to  volume  elements  .         .         .         .         .59 

36.  Poisson's  Equation  in  the  integral  form         .         .         .         .62 

37.  The  average  value  of  the  potential  function  on  a  spherical 

surface 64 

38.  The   equilibrium   of    fluids    at    rest    under  the   action   of 

given  forces 66 

CHAPTER   III. 

THE  NEWTONIAN   POTENTIAL   FUNCTION    IN    THE    CASE   OF 
REPULSION. 

39.  Repulsion  according  to  the  '•  Law  of  Nature  "       .         .         .72 

40.  The  force  at  any  pomt  due  to  a  given  disti'ibution  of  repel- 

ling matter     .........     73 

41.  The  potential  function  due  to  repelling  matter  as  a  measure 

of  work     ^ .  .75 

42.  Gauss's  Theorem  in  the  case  of  repelling  matter  .  .  .75 

43.  Poisson's  Equation  in  the  case  of  repelling  matter  .  .     76 

44.  The  coexistence  of  two  kinds  of  active  matter     .  .  .77 

CHAPTER  IV. 

THE  PROPERTIES  OF  SURFACE  DISTRIBUTIONS.    GREEN'S 
THEOREM. 

45.  The  force  due  to  a  closed  shell  of  repelling  matter        .         .     80 

46.  The  potential  function  is  finite  at  points  in  a  surface  distri- 

bution of  matter    ........     82 

47.  The  normal  force  at  any  point  of  a  surface  distribution        .     85 

48.  Green's  Tlieoreni  .........     87 

49.  Special  cases  under  Green's  Theorem    .         .         .         .         .91 

50.  Surface  distributions  which  are  equivalent  to  certain  vol- 

ume distributions  ........     95 

51.  Those  characteristics  of  the  potential  function  which  are 

sufficient  to  determine  the  function         .         .         .         .96 

52.  Thomson's  Theorem.     Diriclilet's  Principle  .         .         .98 


TABLE  OF   CONTENTS. 


CHAPTER  V. 

ELECTROSTATICS. 
Section  Page 

53.  Introductory 103 

54.  The  charges  on  conductors  are  superficial  ....     104 

55.  General  principles  which  follow  directly  from  the  theory 

of  the  Newtonian  potential  function     ....     106 

56.  Tubes  of  force  and  their  properties 108 

57.  Hollow  conductors 110 

58.  The  charge  induced  on  a  conductor  which  is  put  to  earth  .     114 

59.  Coefficients  of  induction  and  capacity         .         .         .         .115 

60.  The  distribution  of  electricity  on  a  spherical  conductor      .     117 

61.  The  distribution  of  a  given  charge  on  an  ellipsoidal  con- 

ductor   118 

62.  Spherical  condensers 119 

63.  Condensers  made  of  two  parallel  conducting  plates    .         .     122 

64.  The  capacity  of  a  long  cylinder  surrounded  by  a  concentric 

cylindrical  shell .124 

65.  Specific  inductive  capacity .......     125 

66.  The  charge  induced  on  a  conducting  sphere  by  a  charge 

at  an'  outside  point        ,         .         .         .         .         .         .130 

67.  The  energy  of  charged  conductors f34 


THE 


NEWTONIAN  POTENTIAL  FUNCTION. 


3j<Ko 


CHAPTER   I. 

THE  ATTKAOTION  OP  GRAVITATION. 

1.  The  Law  of  Gravitation.  Every  body  in  the  universe 
attracts  every  other  body  with  a  force  which  depends  for  mag- 
nitude and  direction  upon  the  masses  of  the  two  bodies  and 
upon  their  relative  positions. 

An  apj)roximate  vahie  of  the  attraction  between  any  two  rigid 
bodies  may  be  obtained  by  imagining  the  bodies  to  be  divided 
into  small  particles,  and  assuming  that  every  particle  of  the  one 
body  attracts  every  particle  of  the  other  with  a  force  directly 
proportional  to  the  product  of  the  masses  of  the  two  particles, 
and  inversely  proportional  to  the  square  of  the  distance  between 
their  centres  or  other  corresponding  points.  The  true  value  of 
the  attraction  is  the  limit  approached  by  this  approximate  value 
as  the  particles  into  which  the  bodies  are  supposed  to  be  divided 
are  made  smaller  and  smaller. 

2.  The  Attraction  at  a  Point.  By  "the  attraction  at  any 
point  P  in  space,  due  to  one  or  more  attracting  masses."  is 
meant  the  limit  which  would  l)e  approached  by  the  value  of  the 
attraction  on  a  sphere  of  unit  mass  centred  at  P  if  the  radius  of 
the  sphere  were  made  continually  smaller  and  smaller  while  its 
mass  remained  unchanged.  Tlie  attraction  at  P  is,  then,  the 
attraction  on  a  unit  mass  supposed  to  be  concerUrcUed  at  7*. 


2  THE   ATTRACTION   OF    GRAVITATION. 

If  the  attraction  at  every  point  throughout  a  certain  region 
has  a  value  other  than  zero,  the  region  is  called  "a  field  of 
force  "  ;  and  the  attraction  at  any  point  P  in  the  region  is  called 
"  the  strength  of  the  field"  at  that  point. 

3.  The  Unit  of  Force,  It  will  presently  appear  that  all  spheres 
made  of  homogeneous  material  attract  bodies  outside  of  them- 
selves as  if  the  masses  of  the  spheres  were  concentrated  at  their 
middle  points.  If,  then,  k  be  the  force  of  attraction  between 
two  unit  masses  concentrated  at  points  at  the  unit  distance 
apart,  the  attraction  at  a  point  P  due  to  a  homogeneous  sphere 


of  radius  a  and  of  density  p  is  k 


Sr^ 


where  r  is  the  dis- 


tance of  P  from  the  centre  of  the  sphere.  In  all  that  follows, 
however,  we  shall  take  as  our  unit  of  force  the  force  of  attrac- 
tion between  two  unit  masses  concentrated  at  points  at  the  unit 
distance  apart.  Using  these  units,  k  in  the  expression  given 
above  becomes  1,  and  the  attraction  between  two  particles  of 

mass  mi  and  wi2  concentrated  at  points  r  units  apart  is  — ^-?- 

4.  Attraction  due  to  Discrete  Particles.  The  attraction  at  a 
point  P,  due  to  particles  concentrated  at  different  points  in  the 
same  plane  with  P,'  may  be  expressed  in  terms  of  two  com- 
ponents at  right  angles  to  each  other. 


f-r-^^ 


Fig.  1. 


Let  the  straight  lines  joining  P  with  the  different  particles  be 
denoted  by  ?-i,  n,  r^,  •••,  and  the  angles  which  these  lines  make 
with  some  fixed  line  Px  by  aj,  aj,  ag,  •••.     If,  then,  the  masses 


THE   ATTRACTION   OF   GRAVITATION.  3 

of  the  several  particles  are  respectively  wij,  mg,  Wg,  •••,  the 
components  of  the  attraction  at  P  are 

^T-      mjcosai      m^cosag  ,  X^mcosa  r-n 

in  the  direction  Pa;,  and 

Y _  ?»i  sin ny      7)12  sin  ag  _■   ^  ^ .  _ ^^^  sin  a  p„-, 

in  the  direction  P//,  perpendicular  to  Pic. 
The  resultant  force  at  P  is 


P=VX^'+FS  [3] 

and  its  line  of  action  makes  with  Px  the  angle  whose  tangent 
.     Y 

18  —  • 

A" 

If  the  particles  do  not  all  lie  in  the  same  plane  with  P,  we 
ma}-  draw  through  P  three  mutually  perpendicular  axes,  and  call 
the  angles  which  the  lines  joining  P  with  the  different  particles 
make  with  the  first  axis  a^,  a,,  ag,  •••  ;  with  the  second  axis, 
A'  ^21  A'  "'  'i  ^^^  with  the  third  axis,  yi,  yg,  yg,  •••.  The  three 
components  in  the  directions  of  these  axes  of  the  attraction  at 
P  due  to  all  the  particles  are  then 

,-  _"V  ^mcosa  .    -r^_\^m  cos^  ,    y  _\^mcosy         r-.-. 
The  resultant  attraction  is 


and  its  line  of  action  makes  with  the  axes  angles  whose  cosines 

are  respectively 

X     Y  Z 

^\   -,  and  -.  [61 

5.  Attraction  at  a  Point  in  the  Produced  Axis  of  a  Straight 
Wire.  Let  /x  be  the  mass  of  the  unit  of  length  of  a  uniform 
straight  wire  AB  of  length  I,  and  of  cross  section  so  small  that 


4  THE   ATTRACTION   OF   GRAVITATION. 

we  may  suppose  the  mass  of  the  wire  concentrated  in  its  axis 
(see  Fig.  2) ,  and  let  P  be  a  point  in  the  line  AB  produced  at  a 


e--.,^_      ^'^"' 


^  'M' 

^^'   Fig.  2. 


distance  a  from  A.     Divide  the  wire  into  elements  of  length 
Aa;.    The  attraction  at  P  due  to  one  of  these  elements,  M,  whose 

nearest  point  is  at  a  distance  x  from  P,  is  less  than  ^^— -  and 

greater  than  - — ^ -• 

^  {x-]-Axy 

The    attraction   at  P  due   to   the   whole  wire  lies  between 

Z^ —  and    /  — ^ ;  but  these  quantities  approach  the 
ar            Zm^{x  +  Axy 

same  limit  as  Ax  is  made  to  approach  zero,  so  that  the  attrac- 
tion at  P  is 


Ax 


nit  \^fjiAx_  r^+^jxdx^ 


limit     ,    .  _  .  .  _^ 


"1       1 


a      a-\-l 


[7] 


If  the  coordinates  of  P,  A^  and  B  are  respectively  (x,  0,  0) , 
(x'l,  0,  0),  and  {x^  + 1,  0,  0) ,  this  result  may  be  put  into  the  form 

[8] 


6.  Attraction  at  any  Point,  due  to  a  Straight  Wire.  Let  P 
(Fig.  3)  be  any  point  in  tlie  perpendicular  drawn  to  the  straight 
wire  AB  at  A,  and  let  PA  =  c,  AB  =  I,  AM=  x,  and  the  angle 
ABP=  8.  Let  MN  be  one  of  the  equal  elements  of  mass  {fiAx) 
into  which  the  wire  is  divided,  and  call  PiT/,  r.     The  attraction 

at  P  due  to  this  element  is  approximately  equal  to  ^^-^-^  and 

r 

acts  in  some  direction  lying  between  PM  and  PN.    This  attrac- 
tion can  be  resolved  into  two  components  whose  approximate 

values  are  -^ '——  in  the  direction  PA,  and  -^ '——  in  the 

(c^  +  a;-)?  '  (c^+ar^)? 


THE  ATTRACTION   OF   GRAVITATION. 


direction  PL.  The  true  values  of  the  components  in  these 
directions  of  the  attraction  at  P,  due  to  the  whole  wire,  are, 
then,  respectively : 


and 


f    ^"^^     -^r       ^      7:=^  cos  8,  [9] 

X(^.  =  ^[vP^]=^i--«).      [10] 


Fig.  3. 

The  resultant  attraction  is  equal  to  the  square  root  of  the  sum 
of  the  squares  of  these  components,  or  .    , 


/^ 


/^ 


"^H, 


e 


i?  =  c:V2(l-sin8)  =  ^V2(l-cos^P^)  =-^s\n^APB,[\\'\     v: 

and  its  line  of  action  makes  with  PA  an  angle  whose  tangent  is 
1 -sin8__l-cos^P^_  2sin^i^P^ 


cos  8 


sin  APB         2  sin  I  APE  •  cos  |  APB 


tan  ^^Pi?. 


That  is,  the  resultant  attraction  at  P  acts  in  the  direction  of 
the  bisector  of  the  angle  APB. 

From  these  results  we  can  easily  obtain  the  value  of  the 
attraction  at  any  point  P,  due  to  a  uniform  straight  wire  B'B 
(Fig.  4) .  Drop  a  perpendicular  P^l  from  P  upon  the  axis  of 
the  wire.  Let  AB  =  I,  AB'  =  V,  PA  =  c,  ABP  =  8,  .IjB'P  =  8', 
BPB'=6.  The  component  in  the  direction  P-.1  of  the  attrac- 
tion at  P  is  [9] 

u 

-(cos  8  +cos8'), 


THE   ATTRACTION   OF    GRAVITATION. 


and  that  in  the  direction  PL  is 


(sin  8'—  sin  8), 


so  that  the  resultant  attraction  is 
c    ■     I-     ■    -      V     •      /J       c 


i2=^V2[l+cos(8  +  8')]  =  ^cosi(S  +  S')  =  ^sini-^.     [12] 


7^ 


The  line  of  action  PK  ot  R  makes  with  PA  an  angle  <!>  sucli 
that 


tan<^  = 


sin  S'  —  sin  8 
cos  8  +  cos  8' 


tan  i  (8' -8); 


[13] 


and 


.•.5'P^=|-8'+i(S'-8)  =  |-i(8  +  S'), 


^P/r=|-8-i(8'-8)  =  |-i(8+8'). 


It  is  to  be  noticed  that  PK  bisects  the  angle  9,  and  does  not 
in  general  pass  through  the  centre  of  gravity  or  any  other  fixed 
point  of  the  wire.  Indeed,  the  path  of  a  particle  moving  from 
rest  under  the  attraction  of  a  straight  wire  is  generally  curved  ; 
for  if  the  particle  should  start  at  a  point  Q  and  move  a  short 
distance  on  the  bisector  of  the  angle  BQB'  to  Q',  the  attraction 
of  the  wire  would  now  urge  the  particle  in  the  direction  of  the 
bisector  of  the  angle  BQ'B',  and  this  is  usually  not  coincident 
with  the  bisector  of  BQB'. 


THE   ATTRACTION   OF   GRAVITATION.  7 

If  Q*  is  the  area  of  the  cross  section  of  the  wire,  and  p  the 
mass  of  the  unit  volume  of  the  substance  of  which  the  wire  is 
made,  we  may  substitute  for  /a  in  the  formulas  of  this  section 
its  value  qp. 

If  instead  of  a  very  thin  wire  we  had  a  body  in  the  shape  of 
a  prism  or  cylinder  of  considerable  cross  section,  we  might 
divide  this  up  into  a  large  number  of  slender  prisms  and  use  the 
equations  just  obtained  to  find  the  limit  of  the  sum  of  the  attrac- 
tions at  any  point  due  to  all  these  elementary  prisms.  This 
would  be  the  attraction  due  to  the  given  body. 

7.  Attraction  at  a  Point  in  the  Produced  Axis  of  a  Cylinder 
of  Revolution.  In  order  to  find  the  attraction  due  to  a  homo- 
geneous cylinder  of  revolution  at  any  point  P  (Fig.  o)  in  the 
axis  of  the  cylinder  produced,  it  will  be  convenient  to  imagine 
the  cylinder  cut  up  into  discs  of  constant  thickness  Ac,  by 
means  of  planes  perpendicular  to  the  axis. 

Let  p  be  the  mass  of  the  unit  of  volume  of  the  cylinder,  and 
a  the  radius  of  its  base.  Consider  a  disc  whose  nearer  face  is 
at  a  distance  c  from  P,  and  divide  it  into  elements  by  means  of 

b'   b 


Fig.  5. 

radial  planes  drawn  at  angular  inten'als  of  M  and  concentric 
cylindrical  surfaces  at  radial  intervals  of  Ar. 

The  mass  of  any  element  3/ whose  inner  radius  is  r  is  equal 
to  pAc.A^[rAr  +  ^(Ar)-],  and  the  whole  attraction  at  7^  due  to 

M  is  approximately  ^^^gf^^J'^  ?  (^'•)'J  in  a  line  joining  P 

with  some  point  of  3/.     The  component  of  this  attraction  in 
the  direction  PC  is  found  by  multiplying  the  exjxession  just 


THE  ATTRACTION   OF   GRAVITATION. 


given  by 


Vc^^-; 


:,  the  cosine  of  the  angle  CPS,  so  that  the 


attraction  at  P  in  the  direction  PC,  due  to  the  whole  disc,  is 
approximately 

L^  (0^4- 7^)3  Jo      Jo  (c^  +  r^^ 


=  2  7r/3  Ac 


(c2  +  r^)3 


1- 


Vc^+' 


>} 


[14] 


If  the  bases  of  the  cylinder  are  at  distances  Cq  and  Cq  +  h 
from  P,  the  true  value  of  the  attraction  at  P  in  the  direction 
PC^  due  to  the  cylinder  QQ',  is 


limit 
Ac 


nit\^ 


27rpAc 


1- 


_  '     Vc2+ : 


=  27rp  I 


1  — 


dc 


Vc2+  a^. 
=  27rp[7i+V^7+^-V(co  +  /0'+a'].  [15] 

This  is  evidently  the  whole  attraction  at  P  due  to  the  cylin- 
der, for  considerations  of  symmetry  show  us  that  the  resultant 
attraction  at  P  has  no  component  perpendicular  to  PC. 

[14]  gives  the  attraction  due  to  the  elementary  disc  ^P^l'JB', 
on  the  assumption  that  the  whole  matter  of  the  disc  is  coijcen- 
trated  at  the  fape  ABC.  The  actual  attraction  at  P  due  to 
this  disc  may  be  found  by  putting  Co  =  c  and  7i  =  Ac  in  [15]. 

If  ix,  the  radius  of  the  cylinder,  is  very  large  compared  with 
h  and  Co,  the  expression  [15]  for  the  attraction  at  P  due  to  the 
cylinder  approaches  the  value  2Trph. 

8.  Attraction  at  the  Vertex  of  a  Cone.  The  attraction  due  to 
a  homogeneous  cone  of  revolution,  at  a  point  at  the  vertex  of 
the  cone,  may  be  found  by  the  aid  of  [14]. 

If  Fig.  G  represents  a  plane  section  of  the  cone  taken  through 
the  axis,  and  if  PM=  c,  MM'  =  Ac,  and  MB  =  ?•,  the  attraction 
at  P  due  to  the  disc  ABCD  is  approximately 


)Ac 


1  — 


V?T^ 


=  2  7rp  Ac  (1  —  cos  a) , 


THE  ATTRACTION   OF   GRAVITATION. 


imit  ^^£ 


limit 
Ac 


Ac 


and  the  attraction  due  to  the  whole  cone  is 

2  Trp  ( 1  -  cos  a)  Ac  =  2  7r/3  ( 1  -  COS  a)  ^jf^  qT^^ 

=  27rp{l-COSa)  -PL.  [IG] 

*  The  attraction  at  P  due  to  the  frustum  ABKN  is  fouud  by 
subtracting  the  value  of  the  attraction  due  to  the  cone  ABP 
from  the  expression  given  in  [IC].     The  result  is 

27rp(l  -  COSa)  {PL  — PM)  =  2'rrp{\- COSa) ML,         [17] 

and  it  is  easy  to  see  from  this  that  discs  of  equal  thickness  cut 
out  of  a,  cone  of  revolution  at  different  distances  from  the  vertex 
by  planes  perpendicular  to  the  axis  exert  equal  attractions  at 
the  vertex  of  the  cone. 


Fig.  6. 


It  follows  almost  directly  that  the  portions  cut  out  of  two 
concentric  spherical  shells  of  equal  uniform  density  and  equal 
thickness,  bj-  any  conical  surface  having  its  vertex  at  the 
common  centre  P  of  the  shells,  exert  equal  attraction  at  this 
centre ;  but  we  may  prove  this  proposition  otherwise ^  as  fol- 
lows : 

Divide  the  inner  surface  of  the  portion  cut  out  of  one  of  the 
shells  by  the  given  cone  into  elements,  and  make  the  perimeter 
of  each  of  these  surface  elements  the  directrix  of  a  conical 
surface  having  its  vertex  at  P.  Divide  the  given  shells  into 
elementary  shells  of  thickness  Ar  by  means  of  concentric  spheri- 
cal surfaces  drawn  about  P.  In  this  way  the  attracting  masses 
will  be  cut  up  into  volume  elements. 

Let  ML'  (Fig.  7)  represent  one  of  these  elements,  whose 
inner  surface  has  a  radius  equal  to  r ;  then,  if  the  elementary 


10  thp:  attkaction  of  gravitation. 

cone  APB  intercept  an  element  of  area  Aw  from  a  spherical  sur- 
face of  radius  unity  drawn  around  P,  the  area  of  the  surface 
element  at  MM'  is  ?-^Aw,  and  that  at  LL'  is  (r  +  Ar)^Aw.    The 


attraction  at  P  in  the  direction  PM,  due  to  the  element  ML',  is 

approximately 

r^AwAr         .     . 

P 5 —  =  p  AwAr, 

?- 

and  the  component  of  this  in  any  direction  Px,  making  an 

angle  a  with  PJf,  is  approximately  /a  Aw  Ar  cos  a.    The  attraction 

at  P  in  the  du'ectiou  Px,  due  to  the  whole  shell  EDFG,  is, 

then,  ^^ 

X  =  lim  2   p  Ar  Aw  cos  a, 

where  the  sum  is  to  include  all  the  volume  elements  which  go  to 
make  up  the  shell.  If  PF=r^,  PG  =  n,  PF'=:r^,  PG'  =  ry', 
and  fji  =  FG  =  F'G', 

X=   I     \lr  I  COHadoi=p[x  I  COSadw. 

The  attraction  at  P  in  the  same  direction,  due  to  the  shell 
E'D'F'G',  is 

X'  =  p  \     dr  i  cosadw  =  pfx  |  cos  adw. 

But  the  limits  of  integration  with  regard  to  w  are  the  same  in 
both  cases  ;  .•.  X=  X',  which  was  to  be  proved. 

If  tlie  shells  are  of  different  tliicknesses,  it  is  evident  that 
they  will  exert  attractions  at  P  proportional  to  these  thick- 
nesses. 


THE   ATTRACTION   OF   GRAVITATION. 


11 


The  area  of  the  portion  which  a  conical  surface  cuts  out  of  a 
spherical  surface  of  unit  radius  drawn  about  the  vertex  of  the 
cone  is  called  "  the  solid  an^le  "  of  the  conical  surface. 


9.  Attraction  of  a  Spherical  Shell.  In  order  to  find  the 
attraction  at  P,  an}-  point  in  space,  due.  to  a  homogeneous 
spherical  shell  of  radii  Vq  and  ?*i,  it  will  be  best  to  begin  by 
dividing  up  the  shell  into  a  large  number  of  concentric  shells 
of  thickness  Ar,  and  to  consider  first  the  attraction  of  one  of 
these  thin  shells,  whose  inside  radius  shall  be  r. 

Let  p  be  the  density  of  the  given  shell,  that  is,  the  mass  of 
the  unit  of  volume  of  the  material  of  which  the  shell  is  com- 
posed. Join  P  (Fig.  8)  with  0  by  a  straight  line  cutting  the 
inner  surfoce  of  the  thin  shell  at  JV,  and  pass  a  plane  through 
PO  cutting  this  inner  surface  in  a  great  circle  XLSL',  which 


Fig.  8. 

will  serve  as  a  prime  meridian.  L'sing  JV  as  a  pole,  describe 
upon  the  inner  surface  of  the  thin  shell  a  number  of  parallels  of 
latitude  so  as  to  cut  off  equal  arcs  on  XLSL'.  Denote  In'  A.$ 
the  angle  which  each  one  of  these  arcs  subtends  at  0.  Through 
PO  pass  a  number  of  planes  so  as  to  cut  up  each  parallel  of 
latitude  into  equal  arcs.  Denote  by  A(^  the  angle  between  any 
two  contiguous  planes  of  this  series.  By  tliis  means  the  inner 
surface  of  the  elementary  shell  will  be  divided  into  small  quad- 
rilaterals, each  of  which  will  have  two  sides  formed  of  meridian 
arcs,  of  length  r-AO,  and  two  sides  formed  of  arcs  of  parallels 
of  latitude,  of  length  rsin^-A<^  and  rsin(0 +  AO)'A<f>,  where 


12  THE  ATTRACTION   OF    GRAVITATION. 

6  is  the  angle  which  the  radius  drawn  to  the  parallel  of  higher 
latitude  makes  with  ON.  The  area  of  one  of  these  quadri- 
laterals is  approximately  7*^sin(?- A^- At^,  and  the  thickness  of 
the  shell  is  Ar,  so  that  the  element  of  volume  is  approxi- 
mately 7-^sin^- Ar- A^  •  A<^.  Let  PM—y,  tlien  the  attrac- 
tion at  P,  due  to  an  element  of  mass  which  has  a  corner  at 

M,   is   approximately  ^ —,   in   the  direction  PM. 

This  force  ma}-  be  resolved  into  three  components  :  one  in  the 

direction  PO,    the    others  in  directions  perpendicular  to  PO 

and  to  each  other ;    but  it  is  evident  from  considerations  of 

symmetry  that  in  finding  the  attraction  at  P  due  to  the  whole 

shell  we  shall  need  only  that  component  which  acts  in  PO.    This 

.      ,  ,    p?"sin(9- A9-A^A(i-cos7t'P3/  .„  „^ 

IS  approximately  ^ —— ;  or,  it  PO  =  c, 

p?^sin^(c  —  7-cos^)ArA^A<^  p.„-| 

f 

The  attraction  at  P  due  to  the  whole  elementary  shell  is,  then, 
approximately  (truly  on  the  assumption  that  the  whole  mass  of 
the  shell  is. concentrated  at  its  inner  surface), 

Ar  f  fP^  ^"^  0(c-r  cos  0)  (19  d4  ^  ^^  y  ;  [19] 

and  the  true  value  at  P  of  the  attraction  due  to  the  given  shell  is 


Xdr.  [20] 

If  in  the  expression  for  X  we  substitute  for  6  its  value  in 
terms  of  i/,  we  have,  since 

y^  z=z  <r  +  y-  —  2  crcos^, 

and  hence    .  2ydy  =  2crs,\i\6d9, 

Jo  Jy,    2ry-  &  Jy,  \     ly  J 

=g:^'^'-<^+^^  '        [21] 

•f  L     y     At.- 


THE  ATTRACTION   OF   GRAVITATION.  13 

In  order  to  find  the  limits  of  the  integration  with  reg&rd  to  y, 
we  must  distinguish  between  two  cases  : 

I.    If  P  is  a  point  in  the  cavity  enclosed  by  the  given  shell, 
y^  =  r  —  G    and     yj  =  r4-c; 


(?  \_         r-\-c  r  —  c 


=  0,     [22] 


and 


pXdr  =  0;  [23] 

SO  that  a  homogeneous  spherical  shell  exerts  no  attraction  at 
points  in  the  cavity  which  it  encloses. 

II.    If  P  is  a  point  without  the  given  shell, 

2/o  =  c  —  r     and     y^  =  c  +  r  ; 

X  —  ^^ r^— c^  +  (c+^')'  _  y^—  (r+(c  —  r)- 
c^[_  c-]-r  c  —  r 

and  C'Xdr  =  -  ^  (?f  -  r^^) . 

./r„  3  C" 

From  this  it  follows  that  the  attraction  due  to  a  spherical 
shell  of  uniform  density  is  the  same,  at  a  point  without  the  shell, 
as  the  attraction  due  to  a  mass  equal  to  that  of  the  shell  con- 
centrated at  the  shell's  centre. 

If  in  [25]  we  make  ?u=0,  we  have  the  attraction,  due  to  a 
solid  sphere  of  radius  Vi  and  density  p,  at  a  point  outside  the 
sphere  at  a  distance  c  from  the  centre.     This  is 


ATrpr" 

[24] 

[25] 

4  Trpr^ 
3c2 


[26] 


10.  Attraction  due  to  a  Hemisphere.  At  any  point  P  in  the 
plane  of  the  base  of  a  homogeneous  hemisphere,  the  attraction 
of  the  hemisphere  gives  rise  to  two  components,  one  directed 
toward  the  centre  of  the  base,  the  other  peqiendicular  to  the 
plane  of  the  base.  AVe  will  compute  the  values  of  these  com- 
ponents for  the  particular  case  where  P  lies  on  the  rim  of  the 
hemisphere's  base,  and  for  this  purpose  we  will  take  the  origin 


14 


THE  ATTRACTION  OF   GRAyiTATJON. 


of  our  S3'stem  of  polar  coordinates  at  P,  because  by  so  doing 
we  shall  escape  having  to  deal  with  a  quantity  which  becomes 
infinite  at  one  of  the  limits  of  integration.  Denote  the  coordi- 
nates of  any  point  i  in  the  hemisphere  by  r,  $,  ^,  where  (Fig.  9) 
XPN=  </>,    IPL  =  6,    and  PL  =  r. 


Fig.  9. 


If  Ti  be  the  radius  of  the  hemisphere, 
FT  =  PiV^cos  NPT  =  PX  cos  XPN-  cos  NPT  =  2  ri  sin  6  cos  ^. 

cos  XPL 


IK      IK      IL  cos  4>        .    . 

— ■  = =- — ^  =  sin^cos< 

PL        r  r 


ctT>T       P/S       KL      IL  sin  4>        •    n  •    a 

cos  SPL  =  —  = = ^  =  sin^ sm  ^. 

PL        r  r 

The  mass  of  a  polar  element  of  volume  whose  corner  is  at 
L  is  approximately  p-ILA4>- PLM- Ar  or  pr^sin^ArA^A^, 
and  this  divided  by  r^  is  the  attraction  at  P  in  the  direction  PL 
of  the  element,  supposed  concentrated  at  L.  The  components 
of  this  attraction  in  the  direction  PX  and  Pyare  respectively 
psin^ArA^Ac^cosXPi  and  p  sin^ArA^A^  cos^SilL. 

The  component  in  the  direction  Py  of  the  attraction  at  P  due 
to  the  whole  hemisphere  is,  then, 

JTT  y-»n-       ^2  r,  sin  6  COS  (^ 

2d<j>  I  d$  I  psin^^sin</)dr  =  |pri,  [27] 


THE  ATTRACTION   OF   GllAVITATION. 


16 


and  the  component  in  the  direction  Px  is 

XjT  /^n      ^•2ri  Bindco8<^ 

2(Z<^|  d^l  psin-6cos</)dr=|7rpri.  [28] 

This  last  expression  might  have  been  obtained  from  [2G]  by 
making  c  equal  to  r  and  halving  the  result. 

11.  Attraction  of  a  Hemispherical  Hill.  If  at  a  point  on  the 
earth  at  the  southern  extremity  of  a  homogeneous  hemispheri- 
cal hill  of  densit}'  p  and  radius  r^  the  force  of  gravity  due  to  the 
earth,  supposed  spherical,  is  gr,  the  attraction  due  to  the  earth 
and  the  hill  will  give  rise  to  two  components,  g  —  ^pTi  down- 
wards, and  f  Trprj  northwards.  The  resultant  attraction  does 
not  therefore  act  in  the  direction  of  the  centre  of  the  earth,  but 

makes  with  this  direction  an  angle  whose  tangent  is     ^ ""  ' — 


^Ppar, 


Fig.  10. 

Let  4>  (Fig.  10)  be  the  true  latitude  of  the  place  and  (c^  —  a) 
the  apparent  latitude,  as  obtained  by  measuring  the  angle  which 
the  plumb-line  at  the  place  makes  with  the  plane  of  the  equator. 
Let  a  be  the  radius  of  the  earth  and  o-  its  average  density.    Then 


tana=    t^P^i    = ^LPh 

9-ipri      2(7rao--pri) 


[29] 


16  THE   ATTi: ACTION    OF   GRAVITATION. 

The  radius  of  the  earth  is  very  large  compared  with  the 
radius  of  the  hill,  and  a  is  a  small  angle,  so  that  approximately 

a  =  -^-^,  and  the  apparent  latitude  of  the  place  is  <^ ^-^  • 

2acr  2  acr 

If  <f>i  is  the  true  latitude  of  a  place  just  north  of  the  same  hill, 

its  apparent  latitude  will  be  ^^  +  -^-^ ,  and  the  apparent  differ- 

2ao- 

ence  of  latitude  between  the  two  places,  one  just  north  of  the 
hill  and  the  other  just  south  of  it,  will  be  the  true  difference 

plus  ^.     If  there  were  a  hemispherical  cavity  between  the  two 
aa- 

places  instead  of  a  hemispherical  hill,  the  apparent  difference  of 
latitude  would  be  less  than  the  true  difference. 

12.  Ellipsoidal  Homoeoids.  A  shell,  thick  or  thin,  bounded 
by  two  ellipsoidal  surfaces,  concentric,  similar,  and  similarly 
placed,  shall  be  called  an  ellipsoidal  homoeoid. 


Fig.  11. 

It  is  a  property  of  every  such  shell  that  if  any  straight  line 
cut  its  outer  surface  at  the  points  S^S'  (Fig.  11)  and  its  inner 
surface  at  Q,  Q',  so  that  these  four  points  lie  in  the  order 
SQQ'S',  the  length  SQ  will  be  equal  to  the  length  Q'S'. 

We  will  prove  that  the  attraction  of  a  homogeneous  closed 


THE   ATTRACTION   OF  GBAVITATION.  17 

ellipsoidal  homoeoid,  at  any  point  P  in  the  cavity  which  it  shuts 
in,  is  zero. 

Make  P  the  vertex  of  a  slender  conical  surface  of  two 
nappes,  A  and  B,  and  suppose  the  plane  of  the  paper  to  be 
so  chosen  that  PQ  is  the  shortest  and  PM  the  longest  length 
cut  from  any  element  of  the  nappe  A  by  the  inner  surface  of 
the  homceoid.  Draw  about  P  splierical  surfaces  of  radii  PQ, 
PM,  PS,  and  PO,  and  imagine  the  space  between  the  inner- 
most and  outermost  of  these  surfaces  filled  with  matter  of  the 
same  density  as  the  homoioid.  The  nappe  A  cuts  out  a  portion 
from  this  spherical  shell  whose  trace  on  the  plane  of  the 
paper  is  QLOT.  Let  us  call  this,  for  short,  "  the  element 
QLOT."  The  attraction  atP,  due  to  the  element  QMOS  which 
A  cuts  out  of  the  homoeoid,  is  less  than  the  attraction  at  the 
same  point  due  to  the  element  QLOT,  and  greater  than  that 
due  to  the  element  whose  trace  is  KMNS.  But  the  attraction 
at  P,  due  to  the  first  of  these  elements  of  spherical  shells,  is  to 
the  attraction  due  to  the  other  as  the  thickness  of  the  first  shell 
is  to  that  of  the  other,  or  as  QT  \s,  to  KS.  (See  Section  8.) 
The  limit  of  the  ratio  of  QT  to  KS,  as  the  solid  angle  of  the 
cone  is  made  smaller  and  smaller,  is  unity  ;  therefore  the  limit 
of  the  ratio  of  the  attraction  at  P  due  to  the  element  QMOS,  to 
the  attraction  due  to  the  element  of  spherical  shell  whose  trace 
is  QLNS,  is  unit}^  By  a  similar  construction  it  is  eas}'  to  show 
that  the  limit  of  the  ratio  of  the  attraction  at  P,  due  to  the 
element  which  B  cuts  out  of  the  homoeoid,  to  the  attraction  due 
to  the  portion  of  spherical  shell  whose  trace  is  Q'L'N'S',  is 
unity. 

But  the  attractions  at  P,  due  to  the  elements  Q'L'N'S'  and 
QLNS,  are  equal  in  amount  (since  their  thicknesses  are  the 
same)  and  opposite  in  direction,  so  that  if  for  the  elements  of 
the  homoeoid  these  elements  were  substituted,  there  would  be  no 
resultant  attraction  at  P.  In  order  to  get  the  attraction  at  P 
in  any  direction  due  to  the  whole  honKcoid  we  may  cut  up  the 
inner  surface  of  the  homwoid  into  elements,  use  the  perimeter 
of  each  one  of  these  elements  as  the  directrix  of  a  conical  sur- 


18  THE  ATTRACTION   OF   GRAVITATION. 

face  having  its  vertex  at  P,  and  find  the  limit  of  the  sum  of  the 
attractions  due  to  the  elements  which  these  conical  surfaces  cut 
from  the  homcEoid.  Wherever  we  have  to  find  the  finite  limit  of 
the  sum  of  a  series  of  infinitesimal  quantities,  we  may  without 
error  substitute  for  anj-  one  of  these  another  infinitesimal,  tlie 
limit  of  whose  ratio  to  the  first  is  unit}*.  For  the  attractions  at  P 
due  to  the  elements  of  the  homceoid  we  may,  therefore,  substi- 
tute attractions  due  to  elements  of  spherical  shells,  which,  as  we 
have  seen,  destroy  each  other  in  pairs.  Hence  our  proposition. 
A  shell  bounded  by  two  concentric  spherical  surfaces  gives  a 
special  case  under  this  theorem. 

13.  Sphere  of  Variable  Density.  The  density  of  a  homo- 
geneous body  is  the  amount  of  matter  contained  in  the  unit 
volume  of  the  material  of  which  the  body  is  composed,  and  this 
may  be  obtained  by  dividing  the  mass  of  the  bod}^  b}-  its  volume. 

If  the  amount  of  matter  contained  in  a  given  volume  is  not 
the  same  thi'oughout  a  body,  the  body  is  called  heterogeneous, 
and  its  density  is  said  to  be  variable. 

The  average  density  of  a  heterogeneous  body  is  the  ratio  of 
the  mass  of  the  body  to  its  volume.  The  actual  density  p  at 
any  point  Q  inside  the  body  is  defined  to  be  the  limit  of  the 
ratio  of  the  mass  of  a  small  portion  of  the  body  taken  about  Q 
to  the  volume  of  this  portion  as  the  latter  is  made  smaller  and 
smaller. 

The  attraction,  at  any  point  P,  due  to  a  spherical  shell  whose 
density  is  the  same  at  all  points  equidistant  from  the  common 
centre  of  the  spherical  surfaces  which  bound  the  shell  but  dif- 
ferent at  diflferent  distances  from  this  centre,  may  be  obtained 
with  the  help  of  some  of  the  equations  in  Article  9. 

Since  p  is  independent  of  B  and  c^,  it  may  be  taken  out  from 
under  the  signs  of  integration  with  regard  to  these  variables, 
although  it  must  be  left  under  the  sign  of  integration  with  re- 
gard to  ?•. 

Equations  19  to  24  inclusive  hold  for  the  case  that  we 
are  now  considering  as  well  as  for  the  case  when  p  is  constant. 


THE   ATTRACTION   OF    GRAVITATION. 


19 


SO  that  the  attraction  at  all  points  within  the  cavity  enclosed  by 
a  spherical  shell  whose  density  varies  with  the  distance  from  the 
centre  is  zero. 

If  P  is  without  the  shell,  the  attraction  is 


or,  if  p=/(r). 


Xdr  =  )     - 


rdr. 


The  mass  of  the  shell  is  evidently 

limit  V^*"!     „  /*'*» 


[30] 


[31] 


and  [30]  declares  that  a  spherical  shell  whose  density  is  a 
function  of  the  distance  from  its  centre  attracts  at  all  outside 
points  as  if  the  whole  mass  of  the  shell  were  concentrated  at  the 
centre. 

If  ro  =  0,  we  have  the  case  of  a  solid  sphere. 

14.  Attraction  due  to  any  Mass.  In  order  to  find  the  attrac- 
tion at  a  point  P  (Fig.  12),  due  to  any  attracting  masses  J/',  we 
may  choose  a  system  of  rectangular  coordinate  axes  and  divide 


Fio.  12. 


M'  up  into  volume  elements.  If  p  is  the  average  density  of  one 
of  these  elements  (Ar'),  the  mass  of  the  elenu-nt  will  be  p\v' . 
Let  Q,  whose  coordinates  are  x\  y\  z\  be  a  point  of  the  ele- 


20  THE   ATTRACTION   OF   GRAVITATION. 

ment,  find  let  the  coordinates  of  P  be  x,  y,  z.     The  attraction 
at  P  in  the  direction  PQ  due  to  this  element  is  approximately 

— 2?  ^■^cl  the  components  of  this  in  the  direction  of  the  coordi- 

nate  axes  are 

e^cosa',     ^cos^',     and^'cosy',  [32] 

pq-  PQ^  PQ- 

where  a',  y8',  y'  are  the  angles  which  PQ  makes  with  the  positive 
directions  of  the  axes. 
It  is  easy  to  see  that 


,      PL      x'—x 
cos  a'  =  — -  = 


PQ       PQ' 
and,  similarly,  that 

a,      y'—V           A             I      z'  —  z 
cos  B'  =  - — -r ,      and     cos  y'  = 

^       PQ  ^       PQ 

Moreover, 

PQ'  =  PL'  +  LS'  +  W  =  {^'-^y-\-(y'-yr+iz'-z)\ 

and  this  we  wiU  call  ?*^. 

The  true  values  of  the  components  in  the  direction  of  the 
coordinate  axes  of  the  attraction  at  P,  due  to  all  the  elements 
which  go  to  make  up  M',  are,  then, 

y^_   limit  ^pAf'(a;'  — a;) 

_  C  C  C  .  p{x'—x)dx'dy'dz' .         pgg  -. 

-JJJ[{x'-xy+{y'-yf+(z'-zy-^i'         ^     ^^ 

■r7-_   limit  ^ pi^^v' (y' —  y) 

=  CC  C  p(y'-y)dx'dy'dz'  p3    . 

JJJll^x'-xy-+{y'-yy+{z'-zy:\i'        L     "J 

7  _    limit  X^pA?;'(2'— z) 
^  r  r  r p(z'-z)dx'dy'dz' .        i-g^  -, 

JJJlix'- xy-h (y'- yy+  (z'- zy]i'     ^  "- 


THE  ATTRACTION   OF   GRAVITATION.  21 

where  p  is  the  density  at  the  point  {x',  y\  z'),  and  where  the 
integrations  with  regard  to  x\  y',  and  z'  are  to  include  the  whole 
of  M'. 

The  resultant  attraction  at  P,  due  to  M- ,  is 


i2  =  VX2+F2  +  ^';  [34] 

and  its  line  of  action  makes  with  the  coordinate  axes  angles 
whose  cosines  are 

The  component  of  the  attraction  at  the  point  {x,  y,  z)  in  a- 
direction  making  an  angle  c  with  the  line  of  action  of  R  is 
Rcose.    If  the  direction  cosines  of  this  direction  are  A',  yx',  v', 
we  have 

cos€  =  AA'+  .M/x'+  w'. 


15.   The  quantities  X,  Y,  Z,  and  i2,  which  occur  in  the  last 

section,  are  in  general  functions  of  the  coordinates  x,  y,  and  z  of 

the  point  P.    Let  us  consider  X,  whose  value  is  given  in  [33^^] . 

x'  —  X 
If  P  lies  without  the  attracting  mass  3f',  the  quantity  ■ — 

is  finite  for  all  the  elements  into  which  J/'  is  divided.  Let  L 
be  the  largest  value  which  it  can  have  for  anj-  one  of   these 

elements,  then  X  is  less  than  i  I    |   |  pdx'dy'dz',  or  L-M',  and 

this  is  finite.  If  P  is  a  point  within  the  space  which  the  attract- 
ing mass  occupies,  it  is  easy  to  show  that,  whatever  physical 
meaning  we  may  attach  to  X,  it  has  a  finite  value.  To  prove 
this,  make  P  the  origin  of  a  system  of  polar  coordinates,  and 
divide  31'  up  into  elements  like  those  used  in  Section  10.  It 
will  then  be  clear  that 

X=  C  C  Cp sin^Ocos 4, drdddct>,  [3G] 

where  the  limits  are  to  be  chosen  so  as  to  include  all  the  at- 
tracting mass.      Since  sin^^cos<^  can  never  be  greater  than 


22  THE   ATTRACTION   OF   GRAVITATION. 

unit}',  X  is  less  than    III  pdrdOdcf),  which  is  evidently  finite 

when  p  is  finite,  as  it  always  is  in  fact. 
The  corresponding  expressions, 

Y=  C  C  Cpsm-esmcf>drded<f>,  [37] 

and  Z  =  C  C  CpsinOcosedrdedcfi,  [38] 

can  be  proved  finite  in  a  similar  manner ;  and  it  follows  that 
X,  y,  Z,  and  consequently  H,  are  finite  for  all  values  of  x,  y, 
and  z. 

As  a  special  case,  the  attraction  at  a  point  P  within  the  mass 
of  a  homogeneous  spherical  shell,  of  radii  ?o  and  Vi,  and  of  den- 
sity p,  is 

where  r  is  the  distance  of  P  from  the  centre  of  the  shell. 

16.   Attraction  between  Two  Straight  Wires.     Let  AK  and 

BK'  (Fig.  13)  be  two  straight  wires  of  lengths  I  and  I'  and  of 
line-densities   p,   and  p.' ;    and   let   KB  =  c.     Divide  AK  into 

A  f^f^^         K B [1' 

M  M' 

Fig.  13. 


elements  of  length  Ax,  and  consider  one  of  these  MM',  such 
that  AM=x.     The  attraction  of  BK'  on  a  unit  mass  concen- 

'J. 1_~ 

MB     MK 

therefore,  the  whole  element  MM'  whose  mass  is  pAx  were  con 
centrated  at  M,  the  attraction  on  it,  due  to  BK',  would  be 


trated  at  Jf  would  be  (Sections  2  and  5),  p.' 


If, 


p.p.'Ax =  u,p.'Ax 

^^         MB     MK'        ^^        l  +  c-x     l  +  l'-h 


1-      [40] 


THE   ATTKACTION   OF   GRAVITATIO^T. 


23 


The  actual  force,  due  to  the  attraction  of  BK\  with  which  the 
whole  wire  AK  is  urged  toward  the  right,  is 


limit 
Ax 


^i^^\t.'^x  r — ^ ^ — 1 

~^'^X  \x  -  {I  +  V  +  c)~  X  -  {I  +  c)/ 


=  /A/x'  lojr 


x  —  l  —  V  —  c 
x  —  l  —  c 


'=^;x'log(i±^K^^±^.      [41] 


17.  Attraction  between  Two  Spheres.  Consider  two  homo- 
geneous spheres  of  masses  M  and  J/' (F'ig.  14),  whose  centres 
C  and  C  are  at  a  distance  c  from  each  other.  Divide  the  sphere 
M'  into  elements  in  the  manner  described  in  Section  9.  The 
attraction  due  to  M  at  any  point  P'  outside  of  this  sphere  is,  as 
M 


we  have  seen,          ^,  and  its  line  of  action  is  in  the  direction 


P'C. 


Fro.  14. 

Let  P'=(r,  0,<l))  be  any  point  in  tlie  sphere  3/',  and  let 
CP'  =  y.  The  attraction  of  3/  in  the  direction  P'C  on  an 
element  of  mass  pr  sin^Ar  A^A<^  supposed  concentrated  at  P  is 

— ^ ,  and  the  component  of  this  parallel  to  the 

if 
line  C'C  is    ^/•p>-^sin^(c-rcos^)ArA^A<^^      ^^^    ^^^^^   ^.^^ 

f 


24  THE   ATTRACTION   OF   GRAVITATION. 

which  the  whole  sphere  M'  is  urged  toward  the  right  by  the 
attraction  of  M  is,  then, 

W  C  C  rp^s^Q^^^<^^^^0(c  —  ^•cos^)  [-.„-, 

where  the  integration  is  to  be  extended  to  all  the  elements 
which  go  to  make  up  M'.     It  is  proved  in  Section  9  that  the 

M' 

value  of  this  triple  integral  is  —^,  so  that  the  force  of  attraction 

C" 

MM' 

betweea  the  two  spheres  is  — - — 


18.  Attraction  between  any  Two  Rigid  Bodies.  In  order  to 
find  the  force  with  which  a  rigid  body  M  is  pulled  in  any  direc- 
tion (as  for  instance  in  that  of  the  axis  of  a;)  by  the  attraction 
of  another  body  M',  we  must  in  general  find  the  value  of  a 
sextuple  integral. 

Let  31  be  divided  up  into  small  portions,  and  let  Am  be  the 
mass  of  one  of  these  elements  which  contains  the  point  (x,  ?/,  z) . 

The  component  in  the  direction  of  the  axis  of  x  of  the  attrac- 
tion at  (a;,  y,  z)  due  to  M'  is 


///i 


p(x'—  x)dx'dy'dz' 
[(x'-xy+{y'-yy-\-{z'-zyy 


and  this  would  be  the  actual  attraction  in  this  direction  on  a 
unit  mass  supposed  concentrated  at  (cc,  y,  z).  If  the  mass  Am 
were  concentrated  at  this  point,  the  attraction  on  it  in  the  direc- 
tion of  the  axis  of  x  would  be 

.      r  r  r p(x'—x)dx'dy'dz' |- . „-| 

VJj[{x'-xr+iy'-yy+(z'-zy]f  ^     ^ 

The  actual  attraction  in  the  direction  of  the  axis  of  x  of  M' 
upon  the  whole  of  Jf  is,  then, 

limit  V Am  .  r  r  C  p(x'-x)dx'dy'dz'  p^^n 

a;«  =  0^        J  JJl(^a>'-xy-\-{y'-yy+{z'-zy^i'    ^     "^ 


THE   ATTRACTION   OF   GRAVITATION.  25 

If  p'  is  the  density  at  the  point  (x,  y,  z) ,  and  if  the  elements 
into  which  3f  is  divided  are  rectangular  parallelopipeds  of  di- 
mensions Ax,  Ay,  and  Az,  the  expression  just  given  ma}-  be 
written 

C  C  C  C  C  C     p'p(^'-^)(^^ (^^y ^^ dx'dy'dz'  p..-, 

J  J  J  J  J  J  l(x'-xy+{y'-yy-{.{z'-zr^^      ^  ''^ 

where  the  integrations  are  first  to  be  extended  over  31'  and 
then  over  M. 


EXAMPLES. 

1.  Find  the  resultant  attraction,  at  the  origin  of  a  s^ystem  of 
rectangular  coordinates,  due  to  masses  of  12,  IG,  and  20  units 
respectively,  concentrated  at  the  points  (3,  4),  (  —  5,  12),  and 
(8,  —6).     What  is  its  line  of  action  ? 

2.  Find  the  value,  at  the  origin  of  a  system  of  rectangular 
coordinates,  of  the  attraction  due  to  three  equal  spheres,  each  of 
mass  m,  whose  centres  are  at  the  points  (a,  0,  0),  (0,6,0), 
(0,  0,c).  Find  also  the  direction-cosines  of  the  line  of  action 
of  this  resultant  attraction. 

,  3.  Show  that  the  attraction,  due  to  a  uniform  wire  bent  into 
the  form  of  the  arc  of  a  circumference,  is  the  same  at  the  centre 
of  the  circumference  as  the  attraction  due  to  any  uniform 
straight  wire  of  the  same  density  which  is  tangent  to  the  given 
wire,  and  is  terminated  by  the  bounding  radii  (when  produced) 
of  the  given  wire. 

4.  Show  that  in  the  case  of  an  oblique  cone  whose  base  is 
any  plane  figure  the  attraction  at  the  vertex  of  the  cone  due  to 
any  frustum  varies,  other  things  being  equal,  as  the  thickness 
of  the  frustum. 

5.  Find  the  equation  of  a  family  of  surfaces  over  each  one  of 
which  the  resultant  force  of  attraction  due  to  a  uniform  straight 
wire  is  constant. 

6.  Using  the  foot-pound-second  system  of  fundamental  units, 
and  assuming  that  the  average  density  of  the  earth  is  o.G,  com- 
pare with  the  poundal  the  unit  of  force  used  in  this  chapter. 


26  THE  ATTIIACTION   OF   GRAVITATION. 

'  7.  If  in  Fig.  2  we  suppose  P  moved  up  to  A^  the  attraction 
at  P  becomes  infinite  according  to  [7],  and  yet  Section  15 
asserts  that  the  value,  at  any  point  inside  a  given  mass,  of  the 
attraction  due  to  this  mass  is  always  finite.  Explain  this. 
»  8.  A  spherical  cavity'  whose  radius  is  r  is  made  in  a  uniform 
sphere  of  radius  2  r  and  mass  m  in  such  a  way  that  the  centre 
of  the  sphere  lies  on  the  wall  of  the  cavity.  *I<'ind  the  attraction 
due  to  the  resulting  solid  at  different  points  on  the  line  joining 
the  centre  of  the  sphere  with  tlie  centre  of  the  cavity. 

9.  A  uniform  sphere  of  mass  m  is  divided  into  halves  by  the 
plane  AB  passed  through  its  centre  C.  Find  the  value  of  the 
attraction  due  to  each  of  these  hemispheres  at  P,  a  point  on  the 
perpendicular  erected  to  AB  at  (7,  if  CP  =  a. 

10.  Considering  the  earth  a  sphere  whose  density  varies  only 
with  the  distance  from  the  centre,  what  may  we  infer  about  the 
law  of  change  of  this  density  if  a  pendulum  swing  with  the  same 
period  on  the  surface  of  the  earth  and  at  the  bottom  of  a  deep 
mine  ?     What  if  the  force  of  attraction  increases  with  the  depth 

at  the  rate  of  -th  of  a  dyne  per  centimetre  of  descent  ? 
n 

11.  The  attraction  due  to  a  cylindrical  tube  of  length  li  and 
of  radii  Rq  and  i?i,  at  a  point  in  the  axis,  at  a  distance  Cq  from 
the  plane  of  the  nearer  end,  is 


2Trp[Vco2+i2i^-Vco^+i?o''+V(c„-f/0'+iV-V(Co+/0'+i?i']. 

[Stone.] 
12.  A  spherical  cavity  of  radius  h  is  hollowed  out  in  a  sphere 
of  radius  a  and  density  p,  and  then  completely  filled  with 
matter,  of  density  p^.  If  c  is  the  distance  between  the  centre 
of  the  cavity  and  the  centre  of  the  sphere,  the  attraction  due 
to  the  composite  solid  at  a  point  in  the  line  joining  these  two 
centres,  at  a  distance  d  from  the  centre  of  the  sphere,  is 

J     13.  The  centre  of  a  sphere  of  aluminum  of  radius  10  and  of 
density  2.5,  is  at  the  distance  100  from  a  sphere  of  the  same 


4 


THE  ATTRACTION   OF   GRAVITATION.  27 

size  made  of  gold,  of  density  19.  Show  that  the  attraction 
due  to  these  spheres  is  nothing  at  a  point  between  them,  at  a 
distance  of  about  26.6  from  the  centre  of  the  ahiminum  sphere. 
/  [Stone.] 

14.  Show  that  the  attraction  at  the  centre  of  a  sphere  of  radius 
r,  from  which  a  piece  has  been  cut  by  a  cone  of  revolution 
whose  vertex  is  at  the  centre,  is  Trpr  sin-a,  where  a  is  the 
half  angle  of  the  cone.  [Stone.] 

15.  An  iron  sphere  of  radius  10  and  density  7  has  an  eccentric 
spherical  cavity  of  radius  6,  whose  centre  is  at  a  distance  3 
from  the  centre  of  the  sphere.  Find  the  attraction  due  to 
this  solid  at  a  point  25  units  from  the  centre  of  the  sphere, 
and  so  situated  that  the  line  joining  it  with  this  centre  makes 
an  angle  of  45°  with  the  line  joining  the  centre  of  the  sphere 
and  the  centre  of  the  cavit}'.  [Stone.] 

'16.  If  the  piece  of  a  spherical  shell  of  radii  Tq  and  rj,  inter- 
cepted by  a  cone  of  revolution  whose  solid  angle  is  w  and  whose 
vertex  is  the  centre  of  the  shell,  be  cut  out  and  removed,  find 
the  attraction  of  the  remainder  of  the  shell  at  a  point  P  situated 
in  the  axis  of  the  cone  at  a  given  distance  from  the  centre  of 
the  sphere.  If  in  the  vertical  shaft  of  a  mine  a  pendulum  be 
swung,  is  there  any  appreciable  error  in  assuming  that  the  only 
matter  whose  attraction  influences  the  pendulum  lies  nearer  the 
centre  of  the  earth,  supposed  spherical,  than  the  penduluuj 
does  ? 

17.  Show  that  the  attraction  of  a  spherical  segment  is,  at  its 
vertex, 


^M^-Uv 


where  a  is  the  radius  of  the  sphere  and  h  the  height  of  the 
segment. . 

'^   18.    Show  that  the  resultant  attraction  of  a  spherical  segment 
on  a  particle  at  the  centre  of  its  base  is 

_2_5^  [Sa'-S  oh  4-  fr-  (2  a  -  h)  i /ii] . 
3(a  — /i)- 


28  THE   ATTRACTION   OF   GRAVITATION. 

'  19.  Show  that  the  attraction  at  the  focus  of  a  segment  of  a 
paraboloid  of  revolution  bounded  by  a  plane  perpendicular  to 
the  axis  at  a  distance  6  from  the  vertex  is  of  the  form 

A        1      ci  +  b 
4  irpa  log  — ■ — 

■^  20.  Show  that  the  attraction  of  the  oblate  spheroid  formed 
by  the  revolution  of  the  ellipse  of  semiaxes  a,  &,  and  eccen- 
tricity e,  is,  at  the  pole  of  the  spheroid, 

47rp&  (  ,       (l-e')h    .  _i    ■) 

— r-  T  1  ~"  ■^ ^^^  ^  r  ? 

e-     (  e  ) 

and  that  the  attraction  due  to  the  corresponding  prolate  spheroid 
is,  at  its  pole, 

e'  l2e     "  1-e         3 

^  21.  Show  that  the  attraction  at  the  point  (c,  0,  0),  due  to 
the  homogeneous  solid  bounded  b}'  the  planes  x  =  a,  x  =  b,  and 
by  the  surface  generated  by  the  revolution  about  the  axis  of  x 
of  the  curve  y  =f{x) ,  is 

-i  22.  Prove  that  the  attraction  of  a  uniform  lamina  in  the  form 
of  a  rectangle,  at  a  point  P  in  the  straight  line  drawn  through 
the  centre  of  the  lamina  at  right  angles  to  its  plane,  is 

4/i,sm  ^ — z=^ — » 

where  2  a  and  26  are  the  dimensions  of  the  lamina  and  c  the 
distance  of  P  from  its  plane.  [See  Todhunter's  Analytical 
Statics.'\ 


THE  NEWTONIAN   POTENTIAL  FUNCTION.  29 


CHAPTER   II. 

THE  NEWTONIAN  POTENTIAL  FUNCTION  IN   THE   CASE 
or  GRAVITATION. 

19.  Definition.  If  we  imagine  an  attracting  body  M  to  be 
cut  up  into  small  elements,  and  add  togetbei"  all  tbe  fractions 
formed  by  dividing  tbe  mass  of  eacb  element  by  tbe  distance  of 
one  of  its  points  from  a  given  point  P  in  space,  the  limit  of  tbis 
sum,  as  tbe  elements  are  made  smaller  and  smaller,  is  called  tbe 
value  at  P  of  "  tbe  potential  function  due  to  J/." 

If  we  call  tbis  quantity  F,  we  bave 


limit  ^Am  r,--, 


Am 

wbere  Am  is  tbe  mass  of  one  of  tbe  elements  and  r  its  distance 
from  P,  and  wbere  tbe  summation  is  to  include  all  tbe  elements 
wbicb  go  to  make  up  3f. 

If  we  denote  by  p  tbe  average  density  of  the  element  whose 
mass  is  A?)t,  and  call  tbe  coordinates  of  tbe  corner  of  tbis  ele- 
ment nearest  tbe  origin  x',  y\  2',  and  those  of  P,  x^  y,  z,  we  may 

write 

Am  =  p^x'\y'\z\ 
and 

V^CC  C pdx'dy'dz' .     , 

JJJl(^x'-xY+{y'-yy+{z'-zr-]\'  ^     ^ 

where  p  is  tbe  density  at  tbe  point  {x\  y',  z') ,  and  where  the 
triple  integration  is  to  include  tbe  whole  of  the  attracting  mass  M. 

As  tbe  position  of  tbe  point  P  changes,  the  value  of  the  quan- 
tity under  tlie  integral  signs  in  [47]  changes,  and  in  general  V 
is  a  function  of  the  three  space  coordinates,  i.e.,  V=f{x,y,z). 

To  avoid  circumlocution,  a  point  at  which  the  value  of  the 


30  THE  NEWTONIAN  POTENTIAL.  FUNCTION 

potential  function  is  Vq  is  sometimes  said  to  be  "  at  potential 
Vq."  From  the  definition  of  Fit  is  evident  that  if  the  value  at 
a  point  P  of  the  potential  function  due  to  a  system  of  masses 
Ml  existing  alone  is  Vi,  and  if  the  value  at  the  same  point  of 
the  potential  function  due  to  another  system  of  masses  Mo  exist- 
ing alone  is  V^,  the  value  at  P  of  the  potential  function  due  to 
Ml  and  Mo  existing  together  is  F=  Fi  +  F^. 

20.  The  Derivatives  of  the  Potential  Function.     If  P  is  a 

point  outside  the  attracting  mass,  the  quantity 


which  enters  into  the  expression  for  V  in  [47],  can  never  be 
zero,  and  the  quantity  under  the  integral  signs  is  finite  every- 
wliere  within  the  limits  of  integration  ;  now,  since  these  limits 
depend  only  upon  the  shape  and  position  of  the  attracting  mass 
and  have  nothing  to  do  with  the  coordinates  of  P,  we  may  dif- 
ferentiate F  with  respect  to  either  x,  y,  or  z  by  differentiating 
under  the  integral  signs.     Thus  : 


=SSf 


dx'dy'dz'' 


p(x'—x)dx'dy'dz'  p.^.-, 

[{x'-xy+{y'-yy+{z'-zy]i'       L     J 

where  the  limits  of  integration  are  unchanged  by  the  differen- 
tiation. The  dexter  integral  in  this  equation  is  (Section  14) 
the  value  of  the  component  parallel  to  the  axis  of  x  of  the 
attraction  at  P  due  to  the  given  masses,  so  that  we  may  write, 
using  our  old  notation, 

AF=X,  [49] 

and,  similarly,  DyV=Y,  [50] 

D,V=Z.  [51] 

The  resultant  attraction  at  P  is 

E=Vx'  +  Y'+z'=-y{D,vy+{D,vy  +  {D^vy,  [52] 


COS 


IX  THE  CASE   OF   GRAVITATION.  31 

and  the  direction-cosines  of  its  line  of  action  are : 

a  =  -^,    cos/3  =  -^,    and  cosy  =  -^.        [o3] 

It  is  evident  from  the  definition  of  the  potential  function  that 
the  value  of  the  latter  at  any  point  is  independent  of  the  par- 
ticular system  of  rectangular  axes  chosen.  If,  then,  we  wish  to 
find  the  component,  in  the  direction  of  any  line,  of  the  attraction 
at  any  point  P,  we  may  choose  one  of  our  coordinate  axes 
parallel  to  this  line,  and,  after  computing  the  general  value  of 
V,  we  may  differentiate  the  latter  partially  with  respect  to  the 
coordinate  measured  on  the  axis  in  question,  and  substitute  in 
the  result  the  coordinates  of  P. 

21.  Theorem.  The  results  of  the  last  section  may  be  summed 
up  in  the  words  of  the  following 

THEOREM. 

To  find  the  component  at  a  point  P,  in  any  direction  PK,  of 
the  attraction  due  to  any  attracting  mass  M,  tee  may  divide  the 
difference  between  the  values  of  the  potential  function  due  to  M  at 
P  {a  point  beticeen  P  and  K  on  the  straight  line  PK)  and  at  P 
by  the  distance  PP'.  The  limit  approached  by  this  fraction  as 
P'  approaches  P  is  the  component  required. 

"VVe  might  have  arrived  at  this  theorem  in  the  following  wa^-  : 
If  X,  I",  Z  are  the  components  parallel  to  the  coordinate  axes 

of  the  attraction  at  any  point  P,  the  component  in  any  direction 

P/i"  whose  direction-cosines  are  A,  /x,  and  v,  is 

kX  +fjLV+yZ=  XD,  V+  fJiD^  V+  .'/),  V.  [54] 

Let  X,  y,  z  be  the  coordinates  of  /*,  and  x  -f  Ax,  y  ■+-  Ay, 
z-\-Az  those  of  P',  a  neighboring  point  on  the  line  PK. 

If  V  and  V  are  the  values  of  the  potential  function  at  P  and 
P' respectively,  we  have,  by  Taylor's  Theorem, 

V'=V-hAx-DJ^+Ay-D,V+AZ'D,V+e, 

where  £  is  an  infinitesimal  of  an  order  higher  than  the  first. 


32 


THE  NEWTONIAN  POTENTIAL  FUNCTION 


'''^=^-^-r+^,i>.r+^,.D.r+-^,;  m 


n 


XD,V-hf^D,V+vD,V, 


[56] 


jjpi        pp,  ppi       y       ■  pp,       '■     '   pp 

but  Aa;  =  A-PP',   ^y  =  iL'PP\   ^.z  =  vPP\ 

therefore,  ^^Ti,(i^ 

and  this  (see  [54])  is  the  component  in  the  direction  PK  of 
the  attraction  at  P :  whicli  was  to  be  proved. 

22.  The  Potential  Function  everjrwhere  Finite.     If  P  is  a 

point  within  the  attracting  mass,  the  sum  whose  limit  expresses 
the  value  of  the  potential  function  at  P  contains  one  apparently 
infinite  term.  That  V  is  not  infinite  in  this  case  is  easily 
proved  by  making  P  the  origin  of  a  system  of  polar  coordinates 
as  in  Section  15,  when  it  will  appear  that  the  value  of  the 
potential  function  at  P  can  be  expressed  in  tlie  form 


Vp=  C  C  CprB\nedrdedfi>; 


[57] 


and  this  is  evidently  finite. 

Although  Vp  is  everywhere  finite,  \e.i  when  we  express  its 
value  by  means  of  the  equation  [47],  tlie  quantity  under  the  in- 
tegral signs  becomes  infinite  within   the  limits  of  integration, 


z 

^ 

) 

M 

\ 

x' 

1 T 

"M             X 

Fig.  15. 


when  P  is  a  point  inside  the  attracting  mass.  Under  these  cir- 
cumstances we  cannot  assume  without  further  proof  that  the 
result  obtained  by  differentiating  witli  respect  to  x  under  the 
integral  signs  is  really  D^V.     It  is  therefore  desirable  to  com- 


IN  THE  CASE  OF  GRAVITATION.  33 

pute  the  limit  of  the  ratio  of  the  difference  (A,F)  between  the 
values  of  V  at  the  points  P'=(a;4-Aa;,  y,  z)  and  P=(x,  y,  2), 
both  within  the  attracting  mass,  to  the  distance  (Aa;)  between 
these  points.  For  convenience,  draw  through  P  (Fig.  15)  three 
lines  parallel  to  the  coordinate  axes,  and  let  Q  =  {x',  y',  z'). 

Let  PQ  =  r,   P'Q  =  r',    and  X'PQ  =  i/.. 

Then 

r"^  =  9-^4-(Aa;)-—  2  r  •  Ace  •  cos  i/^, 

where        cos  ij/  = -, 


and  ^^rrr(l_r\pdx'dy'dz> 

Ax      J  J  J  \7-'      rj        Ax 


pdx'dy'dz' 
r' r^ -\- rr'- )        Ax 

^2rAa;cosi/^  —(Ax)^  pdx'dy'dz' 
r'  r^  +  rr''^         J        Ax 
Therefore 

j\  TT limit    f^rV 


=///e 


-SIP 

-Iff 


— ——■  •  p  dx'  dy'  dz' 


2 


pdx'  dy'  dz'  cos  if/  r-  -  qt 
^-^ L^^J 

This  last  integral  is  evidently  the  component  parallel  to  the 
axis  of  X  of  the  attraction  at  P,  so  that  the  theorem  of  Article 
21  may  be  extended  to  points  within  the  attracting  mass. 

It  is  to  be  noticed  that  p  is  a  function  of  x',  y',  and  z',  but  not 
a  function  of  x,  y,  and  z,  and  that  we  have  really  proved  that  the 
derivatives  with  regard  to  x,  y,  and  2  of 


fff^^^f''\l=o'dy'dz; 


34 


THE   NEWTONIAN   POTENTIAL   FUNCTION 


where  i''  is  any  finite,  continuous,  and  single-valued  function  of 
x',  y\  and  z',  can  always  be  found  by  differentiating  under  the 
integral  signs,  whether  {x,y,z)  is  contained  within  the  limits  of 
integration  or  not. 

23.  The  Potential  Function  due  to  a  Straight  Wire.  Let 
f/.  be  the  mass  of  the  unit  length  of  a  uniform  straight  wire  AB 
(Fig.  16)  of  length  21.  Take  the  middle  point  of  the  wire  for 
the  origin  of  coordinates,  and  a  line  drawn  perpendicular  to  the 
wire  at  this  point  for  the  axis  of  x. 


Fig.  16. 


The  value  of  the  potential  function  at  any  point  P  {x,  y)  in 
the  coordinate  plane  is,  then,  according  to  [47], 


F, 


=x 


txdy' 


I  [^-^{y'-yn- 


T  =  /^ 


logl^x'+i^y'-yy+y'-yl 

l-y  +  ^/x'+(l-yy 


If  r  =  AP  =  ^/x'+{l-yy,   and    r' =  BP  =  Vx' +  {l  +  yy, 


whence  y 


-r" 


U 


,  we  may  eliminate  x  and  y  from  [59]  and 


express  Vp  in  terms  of  r  and  r'. 
Thus: 

j^         ,      (r-\-2iy-r"-        1     r-\-r'+2l  r^m 

Fp  =  w  log  ^ — ■ =  «.log — ■ — -^ [60] 

It  is  evident  from  [60]  that  if  P  move  so  as  to  keep  the  sum 
of  its  distances  from   the  ends  of  the  wire  constant,  Vp  will 


IN   THE   CASE   OF   GRAVITATION. 


35 


remain  constant.     P's  locus  in  this  case  is  an  ellipse  whose 
foci  are  at  A  and  B. 
From  [59]  we  get 


X 
X 
X 
X 


x" 


a? 


|_r[r  +  (Z-2/)]      r'[r'-(Z  +  2/)]J 

r-(l-y)      /  +  (Z+y)-| 
r  r'         J 


1  —  cos  8—1 


cos  S  +  cos  8'  , 


—  cos  8' 

} 


and  this  (Section  G).  is  the  component  in  the  direction  of  the 
axis  of  X  of  the  attraction  at  P. 

24.  The  Potential  Function  due  to  a  Spherical  Shell.  In 
order  to  find  the  value  at  the  point  P  of  the  potential  function 
due  to  a  homogeneous  spherical  shell  of  density-  p  and  of  radii  Tq 
and  ri,  we  may  make  use  of  the  notation  of  Section  9.    L'^;;^ 

r  r  r  pr^sm$drded(f>  _  r  r  rprd>/drd4> 

If  P  lies  within  the  cavity  enclosed  by  the  shell,  the  limits  of 
y  are  (r  —  c)  and  {r  +  c),  whence 

F=27rp(r,^-V).  [G2] 

If  P  lies  without  the  shell,  the  limits  of  y  are  (c  —  r)  and 
(c  -f-  ^*)  9  whence 

[63] 


.'}  c 


If  P  is  a  point  within  the  mass  of  the  shell  itself,  at  a  dis- 
tance c  from  the  centre,  we  may  divide  the  shell  into  two  parts 


36 


THE   KEWTONIAN   POTENTIAL   FUNCTION 


by  means  of  a  spherical  surface  drawn  concentric  with  the  given 
shell  so  as  to  pass  through  P.  The  value  of  the  potential  func- 
tion at  P  is  the  sum  of  the  components  clue  to  these  portions  of 
the  shell ;  therefore 


o  c 


=  2,,{n^-|} 


4tp     3 


[64] 


If  we  put  these  results  together,  we  shall  have  the  following 
table :  — 


c>n 


ro<c<  o\ 


ri<c 


V= 


2irp{r^-—ri) 
0 
0 


'-^^'■'-i—Tc'-' 


^TTofr^ 


3c 


(9f-r„'^) 


3  \(f 


i£P  (to 
3  c- 


Hn'~ri) 


-+1 


If  we  make  F,  D^F",  and  D^V the  ordinates  of  curves  whose 
abscissas  are  c,  we  get  Fig.  17.* 

Here  LNQS  represents  V,  and  it  is  to  be  noticed  that  this 
curve  is  everywhere  finite,  continuous,  and  continuous  in  direc- 
tion. The  curve  0-45(7  represents  D^V.  This  curve  is  every- 
where finite  and  continuous,  but  its  direction  clianges  abruptly 
when  the  point  P  enters  or  leaves  the  attracting  mass.  The 
three  disconnected  lines  OA,  DE,  and  FG  represent  DJ^V. 

If  the  density  of  the  shell  instead  of  being  uniform  were  a 
function  of  the  distance  from  the  centre  [p  =/(*')  ]i  ^^  should 
have  at  the  point  P,  at  the  distance  c  from  the  centre  of  the 

V  =  ^\f{r).r.dridy.  [65] 


*  See  Thomson  and  Tail's  Treatise  on  Natural  Philosophy. 


IN  THE  CASE   OF   GKAVITATION. 


37 


From  this  it  follows,  as  the  reader  can  easily  prove,  that  the 
value  of  the  potential  function  due  to  a  spherical  shell  whose 
density  is  a  function  of  the  distance  from  the  centre  only  is 


Fig.  17. 

constant  throughout  the  cavity  enclosed  b}'  the  shell,  and  at  all 
outside  points  is  the  same  as  if  the  mass  of  the  shell  were  con- 
centrated at  its  centre. 

25.  Equipotential  Surfaces.  As  we  have  already  seen,  Fis,  in 
general,  a  function  of  the  three  space  coordinates  [l'=/(-P,y,2;)], 
and  in  any  given  case  all  these  points  at  which  the  potential 
function  has  the  particular  value  c  lie  on  the  surface  whose 
equation  is  F  =  /(..,  y,  .)  =  c. 

Such  a  surface  is  called  an  "  equipotential "  or  ''  level  "  sur- 
face. By  giving  to  c  in  succession  dififerent  constant  values, 
the  equation  V=  c  yields  a  whole  family  of  surfaces,  and  it  is 
always  possible  to  draw  through  any  given  point  in  a  field  of 
force  a  surface  at  all  points  of  which  the  potential  function  has 
the  same  value.  The  potential  function  cannot  have  two  differ- 
ent values  at  the  same  point  in  space,  therefore  no  two  different 
surfaces  of  the  family  V=c,  where  Fis  .the  potential  function 
due  to  an  actual  distribution  of  matter,  can  ever  intersect. 


1.05460 


38  THE  NEWTONIAN  POTENTIAL  FUNCTION 

THEOREM. 

If  there  he  any  resultant  force  at  a  point  in  space,  due  to  any 
attracting  masses,  tJiis  force  acts  along  the  normal  to  that  equi- 
potenticd  sujface  on  which  the  point  lies. 

For,  let  V=f{x,  y,z)  =  c  be  the  equation  of  the  equipotential 
surface  drawn  through  the  point  in  question,  and  let  the  coordi- 
nates of  this  point  be  x^,  y^,  Zq.  The  equation  of  the,  plane 
tangent  to  the  surface  at  the  point  is 

(x-Xo)D^,V-^(y-yo)Dy^V+{z-Zo)D,,V=0, 

and  the  direction-cosines  of  any  line  perpendicular  to  this  plane, 
and  hence  of  the  normal  to   the  given  surface  at   the   point 

cos  a  = ^^ ,  [66r\ 

and  cosY= ^^"^^  •  [660] 

But  if  we  denote  the  resultant  force  of  attraction  at  the  point 
(^0?  Voi  ^0)  l>y  -K?  ^^(^  its  components  parallel  to  the  coordinate 
axes  by  X,  Y,   and   Z,   these  cosines  are  evidently    equal  to 

X    Y  Z 

— ,  — ,  and  —  respectively,  so  that  a,  /?,  and  y  are  the  direction- 
R    R  R 

angles  not  only  of  the  normal  to  the  equipotential  surface  at  the 
point  (a^o,  y^,  Zq)  ,  but  also  [35]  of  the  line  of  action  of  the  re- 
sultant force  at  the  point.     Hence  our  theorem. 

Fig.  18  represents  a  meridian  section  of  four  of  the  system 
of  equipotential  surfaces  due  to  two  equal  spheres  whose  sec- 
tions are  here  shaded.  The  value  of  the  potential  function  due 
to  two  spheres,  each  of  mass  M,  at  a  point  distant  respectively 
Ti  and  r^  from  the  centres  of  the  spheres,  is 

vn      ro 


IN   THE   CASE   OF   GRAVITATION. 


39 


and  if  we  give  to  V  in  this  equation  different  constant  values, 
we  shall  have  the  equations  of  different  members  of  the  system 
of  equipotential  surfaces.  Any  one  of  these  surfaces  may  be 
easily  plotted  from  its  equation  by  finding  corresponding  values 


Fig.  18. 


of  9*1  and  To  which  will  satisfy  the  equation  ;  and  then,  with  the 
centres  of  the  two  spheres  as  centres  and  these  values  as  radii, 
describing  two  spherical  surfaces.  The  intersection  of  these 
surfaces,  if  they  intersect  at  all,  will  bo  a  line  on  the  surface 
required. 

If  2(1  is  the  distance  between  the  centres  of  the  spheres, 

2  V 

V  =  —^   gives  an  equipotential  surface  shaped  like  an  hour- 
a 

glass.  Larger  values  of  V  than  this  give  equipotential  sur- 
faces, each  one  of  which  consists  of  two  sei)arate  closed  ovals, 
one  surrounding  one  of  the  spheres,  and  the  otlier  the  other. 

2  3/ 

Values  of  V  less  than give  single  surfaces  which  look  more 

a 

and  more  like  ellipsoids  the  smaller  T''  is. 

Several   diagrams    showing    the    forms  of   the   equipotential 
surfaces  due  to  different  distributions  of  matter  are  given  at 


40  THE  NEWTONIAN   POTENTIAL  FUNCTION 

the  end  of  the  first  volume  of  Maxwell's  Treatise  on  Electricity 
and  Magnetism. 

26.  The  Value  of  V  at  Infinity.     The  value,  at  the  point  P, 

of  the  potential  function  due   to   any  attracting  mass  M  has 

been  defined  to  be 

Y_  limit  "^Am 
*^       A»i=0/  v~^' 

Let  Tq  be  the  distance  of  the  nearest  point  of  the  attracting 
mass  from  P,  then 

F<i'VAm  or  —.  [67] 

M 

The  fraction  —  has  a  constant  numerator,  and  a  denominator 

''"        . 
which  grows  larger  without  limit  the  farther  P  is  removed  from 

the  attracting  masses  ;  hence,  we  see  that,  other  things  being 

equal,  the  value  at  P  of  the  potential  function  is  smaller  the 

farther  P  is  from  the  attracting  matter  ;  and  that  if  P  be  moved 

away  indefinitely,   the  value  of    the    potential    function  at  P 

approaches  zero  as  a  limit.     In  other  words,  the  value  of  the 

potential  function  at  '•'■  infinity "  is  zero. 

27.  The  Potential  Function  as  a  Measure  of  Work.     The 

amount  of  work  required  to  move  a  unit  mass,  concentrated  at 
a  point,  from  one  position,  Pj,  to  another,  Pj,  by  any  path,  in 
face  of  the  attraction  of  a  system  of  masses,  M,  is  equal  to 


Vi  —  V21  where  V\  and  V2  are  the  values  at  Pi  and  P^  of  the 
potential  function  due  to  M. 

To  prove  this,  let  us  divide  the. given  path  into  equal  parts 
of  length  As,   and  call   the  average   force   which   opposes  the 


IN   THE   CASE   OF   GRAVITATION.  41 

motion  of  the  unit  mass  on  its  journey  along  one  of  these 
elements  AB  (Fig.  19),  F.  The  amount  of  work  required  to 
move  the  unit  mass  from  ^1  to  J5  is  FAs,  aud  the  whole  work 
done  by  moving  this  mass  from  Pi  to  P.,  will  be 


limit  "V  ^J  Ti . 


As  As  is  made  smaller  and  smaller,  the  average  force  opposing 
the  motion  along  AB  approaches  more  and  more  nearly  the 
actual  opposing  force  at  A,  which  is  —D^V:  therefore 

,^'"V*  V'FAs  =  -  f^'A  V-  ds  =  F,  -  Fo. 

It  is  to  be  carefully  noticed  that  the  decrease  in  the  potential 
function  in  moving  from  P,  to  P.,  measures  the  work  required 
to  move  the  unit  mass  from  Pi  to  P.,.  If  P.,  is  removed  farther 
and  farther  from  3/,  Vo  approaches  zero,  and  Fj—  T^,  approaches 
Vi  as  its  limit,  so  that  the  value  at  any  point  Pj,  of  the  poten- 
tial function  due  to  any  system  of  attracting  masses,  is  equal 
to  the  work  which  would  be  required  to  move  a  unit  mass,  sup- 
posed concentrated  at  Pj,  from  Pi  to  "  infinity"  by  any  path. 

The  work  (IF)  that  must  be  done  in  order  to  move  an  attract- 
ing mass  M'  against  the  attraction  of  any  other  mass  J/,  from 
a  given  position  by  any  path  to  ''  infinity,"  is  the  sum  of  the 
quantities  of  work  required  to  move  thft  several  elements  (Awi') 
into  which  we  may  divide  M' ,  and  this  may  be  written  in  the 
form 

C  C  C  C  C  C  pp'dxd>/dzdx\h/'dz'  j-^^-. 

^J  J  J  J  J  J  w^)'^!r->'y^+w^^'¥'      ' 

W  is  called  by  some  writers  "the  potential  of  the  mass  M' 
with  reference  to  the  mass  J/"  ;  by  others,  the  negative  of  W  is 
called  '-the  mutual  potential  energy  of  3/ and  3/'." 

Ill   many  of    the   later    books    on    this    subject,    the    word 


42  THE   NEWTONIAN   POTENTIAL   FUNCTION 

"potential"  is  never  used  for  the  value  of  the  potential  func- 
tion at  a  point,  but  is  reserved  to  denote  the  work  required  to 
move  a  mass  from  some  present  position  to  infinitj*.  If  V  is 
the  value  of  the  potential  function  at  a  point  P,  at  which  a 
mass  m  is  supposed  to  be  concentrated,  mV  is  the  j^otential 
of  the  mass  m.  If  we  could  have  a  unit  mass  concentrated 
at  a  point,  the  potential  of  this  mass  and  the  value  of  the  poten- 
tial function  at  the  point  would  be  numerically'  identical, 

28.  Laplace's  Equation.  Wc  have  seen  that  the  value  of  the 
potential  function  and  the  component  in  any  direction  of  the 
attraction  at  the  point  P  are  always  finite  functions  of  the  space 
coordinates,  whether  P  is  inside,  outside,  or  at  the  surface  of 
tlie  attracting  masses.  "We  have  seen  also  that  by  differentiating 
V  at  an}'  point  with  respect  to  an}'^  directioij  we  may  find  the 
always  finite  component  in  that  direction  of  the  attraction  at 
the  point.  It  follows  that  D^V,  DyV,  D^V  are  everywhere 
finite,  and  that,  in  consequence  of  this,  the  potential  function 
is  everywhere  continuous  as  well  as  finite. 

If  P  is  a  point  outside  of  the  attracting  masses,  the  quantity 
under  the  integral  signs  in  [48],  b}'  which  dx'dy'dz'  is  multi- 
l)lied,  cannot  be  infinite  within  the  limits  of  integration,  and  we 
can  find  D/F  by  differentiating  the  expression  for  D^V  under 
the  integral  signs. 

In  this  case 

D^V=  C  C  C^i^'-^y-r"^ dx'dy'dz',  [69] 

and  similarly, 

DfV=  ff  n(y'-yy-^p  dx'dy'dz',  [70] 

i)/  V  =  ffP ^^'~  ^Y  ~  '^ pdx'dTj'dz'.  [71] 

Whence,  for  all  points  exterior  to  the  attracting  masses, 

D/F+I>/F+Z>/F=0.  [72] 

This  is  Laplace's  Equation. 


IX   THE   CASE   OF   GRAVITATION.  43 

The  operator  (/>/  +  Z>/  +  D-)  is  sometimes  denoted  by  the 
symbol  V-,  so  that  [72]  may  be  written 

\-V=0.  [73] 

The  potential  function,  due  to  every  conceivable  distribution 
of  matter,  must  be  such  that  at  all  points  in  empty  space 
Laplace's  PZquation  shall  be  satisfied. 

29.  The  Second  Derivatives  of  the  Potential  Function  are 
Finite  at  Points  within  the  Attracting  Mass.  If  the  point  P 
lies  within  the  attracting  mass,  Fand  D^V  are  finite,  but  the 
quantity  under  the  integi'al  signs  in  the  expression  for  D^V 
becomes  infinite  within  the  limits  of  integration,  and  we  cannot 
assume  that  D^V  may  be  found  by  differentiating  D^V  under 
the  integral  signs.  In  order  to  find  Z^/F"  under  these  circum- 
stances, it  is  convenient  to  transform  the  equation  for  D^V. 
Let  us  choose  our  coordinate  axes  so  as  to  have  all  tlie  attract- 
ing mass  in  the  first  octant,  and  divide  the  projection  of  the 
contour  of  this  mass  on  the  plane  yz  into  elements  {dy'dz'). 
Upon  each  one  of  these  elements  let  us  erect  a  right  prism, 
cutting  the  mass  twice  or  some  other  even  number  of  times. 
Consider  one  of  the  elements  cly'dz'  whose  corner  next  the 
origin  has  the  coordinates  0,  ?/',  and  z'.  The  i)rism  erected  on 
this  element  cuts  out  elements  ds^,  d.%.  ds.^,  ds^,  ■••  ds.-,,^  from  the 
surface  of  the  attracting  mass  and  that  edge  of  the  prism  which 
is  perpendicular  to  the  plane  yz  at  (0,  //',  z')  cuts  into  the 
surface  at  points  whose  distances  from  the  piano  of  yz  are 
«,,  Oj,  Or,,  •••  02n-n  ^"d  o"t  of  the  surface  at  i)oints  whose  dis- 
tances from  the  same  plane  are  a.^,  04  Og,  •••  «.„.  At  every  one 
of  these  points  of  intersection  draw  normals  towards  the  int^ior 
of  the  attracting  mass,  and  call  the  angles  which  tliese  normals 
make  with  the  positive  direction  of  the  axis  of  x,  ai,  a._,,  a^.  ••.  «-,„. 
It  is  to  be  noticed  that  «,,n3,  05,  •••  h2„_i  are  all  acute,  and  that 
Co,  04,  Qg,  ■••  ao„  are  all  obtuse.  The  element  dy'dz' miiy  be  re- 
garded  as    the    common    projection   of   the    surface    elements 


44 


THE   NEWTONIAN  POTENTIAL   FUNCTION 


dsi,  ds2,  dsg,  •••  dson,  and,  so  far  as  absolute  value  is  concerned, 
the  following  equations  hold  approximately  : 

dy'dz'==  dsi  cosaj  =  ds2  cosaa  =  ds^  cosog  =  ••.  =  ds2„  cosa2„. 

But  dy'dz',dSi,ds2,dSs,  etc.,  are  all  positive  areas,  and  cos  a,, 
cosa4,  cosag,  etc.,  are  negative,  so  that,  paying  attention  to 
signs  as  well  as  to  absolute  values,  we  have 

dy'dz'=  -\-dsi  cosai=  —ds^  cosa2=  +dsQ  co3a3=  —ds^  cosa4=  etc. 


Fig.  20. 


Now 


F=J/p("'-">f"'''''''"'=J/dy'&'/pJ.'( -  i) dx',  [74] 


and  in  order  to  find  the  value  of  this  expression  by  the 
use  of  the  prisms  just  described,  we  are  to  cut  each  one 
of  these  prisms  into  elementary  rectangular  parallelopipeds  by 
planes  parallel  to  the  plane  of  yz ;  we  are  to  multiply  the 
values  of  every  one  of  these   elements  which  lies  within  the 

attracting  mass  by  the  value  of  p  DJ  ( ]  at  its  corner  next 

the  origin  [^.e.,  at  (x',y\z')']  ;  and  we  are  to  find  the  limit  of 
the  sum  of  these  as  dx'  is  made  smaller  and  smaller.  We  are 
then  to  compute  a  like  expression  for  each  of  the  other  prisms, 
and  to  find  the  limit  of  the  sum  of  the  whole  as  the  bases  of  the 


IN  THE  CASE  OF   GRAVITATION.  45 

pi-isms  are  made  smaller  and  smaller  and  their  number  eorres-' 
pondingly  increased. 

Wherever  the  function  ^  is  a  continuous  and  finite  function 
of  x',  we  have 


hence,  if  the  elementary  prisms  cut  the  surface  of  the  attracting 
mass  only  twice, 

D,V=ffdy'dz'    -'L\+JJj*l^DJpdx'di/'dz';     [75] 

x'=  fl, 

and,  in  general, 

J  J         L^i     r-2     n     '-4  r,„j 

-h  C  C  f\l)J  p  dx'  dy'  dz'  [76] 

Zl  Pi                         P'>                         Ps 
\  — COSaiCZSj-l -COSa.,dS2-^ COSOgfiSgH 

+  ^'cosa2„c7s2„  j  +  J  J  J  -DJpdx'dy'dz',  [77] 

Pk  P 

where  —  is  the  value  of  the  quantity  -  at  the  point  \Yhere  the 

7\  "   r 

line  y  =  y'i  z  —  2' cuts  the  surface  of  the  attracting  mass  for  the 
A:th  time,  counting  from  the  plane  rjz. 

In  order  to  find  the  value  of  the  limit  of  the  sum  which  occurs 
in  this  expression,  it  is  evident  that  we  may  divide  the  entire  sur- 
face of  the  attracting  mass  into  elements,  multiply  the  area  of  each 

element  by  the  value  of  ^     ^°'  at  one  of  its  points,  and  find  the 

r 

limit  of  the  sum  formed  b}'  adding  all  these  products  together ; 

but  this  is  equivalent  to  the  surface  integral  of  ^- taken  all 

r 
over  the  outside  of  the  attracting  mass,  so  that 

AF=J'^cosacZs  +j^j-^dx'dy'dz\  [78] 


46  THE   NEWTONIAN   POTENTIAL   FUNCTION 

where  the  first  integral  is  to  be  taken  all  over  the  surface  of  the 
attracting  mass  and  the  second  throughout  its  volume.  This 
expression  for  D^Vis  in  some. cases  more  convenient  than  that 
of  [48]. 

We  have  proved  this  transformation  to  be  correct,  however, 

onlv  when  ^  is  finite  throughout  the  attracting  mass.     If  P  is  a 
r 

point  within  the  mass,  ^  is  infinite  at  P.     In  this  case  surround 
r 

P  by  a  spherical  surface  of  radius  e  small  enough  to  make  the 
whole  sphere  enclosed  by  this  surface  lie  entirely  within  the 
attracting  mass.  This  is  possible  unless  P  lies  exactly  upon 
the  surface  of  the  attracting  mass.  Shutting  out  the  little 
sphere,  let  V2  be  the  potential  function  due  to  the  rest  (T^)  of 
the  attracting  mass  :  then,  since  P  is  an  outside  point  with  re- 
gard to  T2,  we  have,  by  [78], 

DxV2=  j  -cosa- ds'-\-  I  -cos ads  +  I    I   I  — ^ — dx'dy'dz',   [79] 

where  the  first  integral  is  to  be  extended  over  the  spherical 
surface,  which  forms  a  pai't  of  the  boundary  of  the  attracting 


Fig.  21. 

mass  to  which  V2  is  due  ;  the  second  integral  is  to  be  taken 
over  all  the  rest  of  the  bounding  surface  of  the  attracting  mass  ; 
and  the  triple  integral  embraces  the  volume  of  all  the  attracting 
mass  which  gives  rise  to  F2. 

As  e  is  made  smaller  and  smaller,  V2  approaches  more  and 
more  nearly  the  potential  function  V,  due  to  all  the  attracting 
mass. 

In  the  integral   I  -  cos  a  ds',  cos  a  can  never  be  greater  than  1 

nor  less  than  —  1 ,  so  that  if  p  is  the  greatest  value  of  p  on  the 


IN  THE  CASE   OF   GRAVITATION.  47 

surface  of  the  sphere,  the  absolute  value  of  the  integral  must  be 
less  than  -J  ds'  or  Airpe,  and  the  limit  of  this  as  e  approaches 

zero  is  zero.     The  second  integral  in  [79]  is  unaltered  by  any 

change  in  e.     If  we  make  P  the  origin  of  a  system  of  polar 

coordinates,  it  is  evident  that  the  triple  integral  in  [79]  may  be 

written  C  C  C     . 

I   j    \D^p'r&va.QdrdOd<^,  [80] 

and  the  limit  which  this  approaches  as  e  is  made  smaller  and 
smaller  is  evidently  finite,  for,  if  r  =  0,  the  quantity  under  the 
integral  sign  is  zero. 
Therefore, 

limit  ^^^^  ^  ^y^  ^P_  ^^^^^^  ^Jffl^ax'dy'dz',  [81] 

and  [79]  is  true  even  when  P  lies  within  the  attracting  mass. 
Under  the  same  conditions  we  have,  similarly, 


and 


D^V=f^-cosfids  -^jff^  dx'dy'dz',  [82] 

D,V=f^co8yds  +fjf^  dx'dy'dz'.  [83] 

Observing  that  in  these  surface  integrals  r  can  never  be  zero, 
since  we  have  excluded  the  case  where  P  lies  on  the  surface  of 
the  attracting  mass,  and  that  the  triple  integrals  belong  to  the 
class  mentioned  in  the  latter  part  of  Section  22,  we  will  differ- 
entiate [81],  [82],  and  [83]  with  respect  to  x,  ?/,  and  z  respcc- 
tiveW,  b^'  differentiating  under  the  integral  signs.  If  the  results 
are  finite,  we  may  consider  the  process  allowable. 

Performing  the  work  indicated,  we  have 

D^'V=fpcosa-Dj^\ls+ffyDA\D:p.dx'dy'dz\[S4] 
DJ'V=  fpcoBft-DAys  +  ffjDJ^\D;p-dx'dy'dz\lSr>^ 
Z>/  F= Jp  cos  y .  A  f-\ls  +ff  fD,  (^-^  •  DJp .  dx'  dy'dz'.  [8(i:\ 


48  THE  NEWTONIAN  POTENTIAL  FUNCTION 

and  by  making  P  the  centre  of  a  system  of  polar  coordinates 
and  transforming  all  the  triple  integrals,  it  is  easy  to  show  that 
the  values  of  D^V^  D^V^  D^V  here  found  are  finite  whether 
P  is  within  or  without  the  attracting  mass.  This  result*  is 
important. 

30.  The  Derivatives  of  the  Potential  Function  at  the  Surface 
of  the  Attracting  Mass.  Let  the  point  P  lie  on  the  surface 
of  the  attracting  mass,  or  at  some  other  point  or  surface  where 
p  is  discontinuous.  Make  P  the  centre  of  a  sphere  of  radius  e, 
and  call  the  piece  which  this  sphere  cuts  out  of  the  attracting 
mass  Ti  and  the  remainder  of  this  mass  71-  Let  Fi  and  V^  be 
the  potential  functions  due  respectively  to  7\  and  T^,  then 

and  the  increment  [A(Z)^F')]  made  in  D^V  by  moving  from  P 
to  a  neighboring  point  P',  inside  Ti,  is  equal  to  the  sum  of  the 
corresponding  increments  [A(Z)^Fi)  and  A(Z>j.F2)]  made  in 
D,Vi  and  D,V2. 

With  reference  to  the  space  Tj,  P  is  an  outside  point,  so  that 
the  values  at  P  of  the  first  derivatives  of  V2  with  respect  to  x, 
y,  and  z  are  continuous  functions  of  the  space  coordinates  and 

p'^™i*oA(AF,)  =  0. 

Let  do)  be  the  solid  angle  of  an  elementary  cone  whose  vertex 
is  at  any  fixed  point  0  in  Ti  used  as  a  centre  of  coordinates. 


Fig.  22. 

The  element  of  mass  will  be  pi^dtadr.     The  component  in  the 
direction  of  the  axis  of  x  of  the  attraction  at  0  due  to  Tj  is  the 

*  Lejeune  Dirichlet,  Vorlesungen  iiher  die  im  umgekehrten  Verhciltniss  des 
Quadrats  der  Entfernung  wirkenden  Krdfte. 

Riemann,  Schwere,  Electricitdt,  und  Magnetismus. 


IX   THE   CASE   OF    GRAVITATION.  49 

limit  of  the  sum  taken  throughout  jTi  of  ^ — —:^ — ,  where  a  is 

/" 

the  cosine  of  the  angle  which  the  line  joining  0  with  the  element 
in  question  makes  with  the  axis  of  x.  The  difference  between 
the  limits  of  o  is  not  greater  than  47r,  and  the  difference  be- 
tween the  limits  of  r  is  not  greater  than  2  c.  If,  then,  k  is  the 
greatest  value  which  pa  has  in  T^, 

It  follows  from  this  that  if  F'  is  a  point  within  7\  so  that 
PP'<  c,  the  cliange  made  in  D^Vi  by  going  from  P  to  P'  is  far 
less  than  ICttkc  ;  but  this  last  quantity  can  be  made  as  small  as 
we  like  by  making  6  small  enough,  so  that 

whence 

limit     A  /  7~K  T7-\  limit     *  /t-,  tt-n    r       limit     .  /  t-v  tt-n       n 

PP,^Q^(D,V)  =  pp/^o  ^(AFi)  +  pp>^o  ^(AFs)  =  0, 

and  2),  V  varies  continuously  in  passing  through  P.  In  a  similar 
manner,  it  may  be  proved  that  D^V  and  D^V  are  everywhere, 
even  at  places  where  the  density  is  discontinuous,  continuous 
functions  of  the  space  coordinates. 

The  results  of  the  work  of  the  last  two  sections  are  well  illus- 
trated 1)3"  Fig.  17.  We  might  prove,  with  "the  help  of  a  trans- 
formation due  to  Clausius,*  that  the  second  derivatives  of  the 
potential  function  are  finite  at  uU  points  on  the  surface  of  the 
attracting  matter  where  the  curvature  is  finite,  but  that  these 
derivatives  generally  change  their  values  abruptly  whenever  the 
point  P  crosses  a  surface  at  which  p  is  discontinuous,  as  at  tlie 
surface  of  the  attracting  masses.  The  fact,  however,  that  this 
last  is  true  in  the  special  case  of  a  homogeneous  spherical  sliell 
suflSces  to  show  that  we  cannot  expect  the  second  derivatives 
of  V  to  have  definite  values  at  the  boundaries  of  attracting 
bodies. 

*  Die  Pot cntiiilf unction  und  das  Potential,  §§  19-24. 


60  THE  NEWTONIAN  POTENTIAL  FUNCTION 

31.  Gauss's  Theorem.  If  any  closed  surface  T  drawn  in  a 
field  of  force  be  divided  up  into  a  large  number  of  surface 
elements,  and  if  each  one  of  these  elements  be  multiplied  hy  the 
component,  in  the  direction  of  the  interior  normals  of  the  force 
of  attraction  at  a  point  of  the  element,  and  if  these  products  be 
added  together,  the  limit  of  the  sum  thus  obtained  is  called  the 
''  surface  integral  of  normal  attraction  over  T." 

If  any  closed  surface  T  be  described  so  as  to  shut  in  com- 
pletely' a  mass  m  concentrated  at  a  point,  the  surface  integral 
of  normal  attraction  due  to  ?3i,  taken  over  T,  is  4ir?n;  and,  in 
general,  if  any  closed  surface  T  be  described  so  as  to  shut  in 
completely  any  system  of  attracting  masses  M,  the  surface  in- 
tegral over  T  of  the  normal  attraction  due  to  M  is  4  tvM. 


Fig.  23. 

In  order  to  prove  this,  divide  T  up  into  surface  elements, 
and  consider  one  of  these  ds  at  Q.     The  attraction  at  Q  in  the 

direction  QO,  due  to  the  mass  m  concentrated  at  0,  is  —^^-  =  — 
The  component  of  this  in  the  direction  of  the  interior  normal  is 

7)1' 

—  cosa,  and  the  contribution  which  ds  yields  to  the  sum  whose 

limit  is  the  surface  integral  required  is  -^ — .      Connect 

every  point  of  the  perimeter  of  ds  with  0  by  a  straight  line, 
thus  forming  a  cone  of  such  size  as  to  cut  out  of  a  spherical 
surface  of  unit  radius  drawn  about  0  an  element  dw,  say.  If  we 
draw  about  0  a  sphere  of  radius  r  =  OQ,  the  cone  will  intercept 
on  its  surface  an  element  equal  to  r^-d(o.     This  element  is  the 


IN   THE   CASE   OF   GRAVITATION.  61 

projection  on  the  spherical  surface  of  ds  ;  hence  dscosa  =  r-dw, 
approximately,  and  the  contribution  of  the  element  ds  to  our 
surface  integral  is  vidi).  But  an  elementary  cone  may  cut  the 
surface  more  than  once  ;  indeed,  any  odd  number  of  times.  Con- 
sider such  a  cone,  one  element  of  which  cuts  the  surface  thrice 
in  Si,  So,  and  S-j.  Let  OSi,  OSo,  and  OSo  be  called  ?-i,  r,,  and 
rg  respectively,  and  let  the  surface  elements  cut  out  of  T  by  the 
cone  be  dsi,  ds.,,  and  dsg,  and  the  angles  between  the  line  S^O 
and  the  interior  normals  to  T  at  S^,  S2,  and  S^  be  aj,  ag,  a,.  It  is 
to  be  noticed  that  when  the  cone  cuts  out  of  T,  the  corresponding 
angle  is  acute,  and  that  when  it  cuts  in,  the  corresponding  angle 
is  obtuse,  oj  and  «3  are  acute,  and  ao  obtuse.  If  we  draw  about 
0  three  spherical  surfixces  with  radii  rj,  ?'2,  and  rg  respectively, 
the  cone  will  cut  out  of  these  the  elements  Vi'do},  r.fdu),  and 
r-idw.  In  absolute  size,  ds^  =  Vidoi  secoi,  ds.,  =  r.?doi  secaj,  and 
ds3=  ?*3^dwsecci3,  approximately,  but  ds.,  and  r^d<a  are  both  posi- 
tive, being  areas,  and  secao  is  negative.  Taking  account  of 
sign,  then,  cZsj  = —J^cZw  sec  ao,  and  the  cone's  three  elements 
3-ield  to  the  surface  integral  of  normal  attraction  the  quantity 

/      dSiQOSa,    ,    d.So  cos  ao    ,    dSoCOSn.\  .J  7       I     7    \  7 

( 7n  — -^ = -| ]  —  m(doi  —  dio  +  dw)  =  m dw. 

\  'V  >V  'V'      / 

However  many  times  the  cone  cuts  T,  it  will  yield  mdw  to  the 
surface  integral  lequired  :  all  such  elementary  cones  will  yield 

then  m  /   dot  =  m47r  if  T  is  closed,  and,  in  general,  m0,  when 

0  is  the  solid  angle  which  T  subtends  at  0. 

If,  instead  of  a  mass  concentrated  at  a  point,  we  have  any 
distrilnition  of  masses,  we  ma}-  divide  these  into  elements,  and 
apply  to  each  element  the  theorem  just  proved  ;  hence  our  gen- 
eral statement. 

If  from  a  point  0  without  a  closed  surface  T  an  elementary 
cone  be  drawn,  the  cone,  if  it  cuts  T  at  all,  will  cut  it  an  even 
number  of  times.  Using  the  notation  just  explnined,  tho  con- 
tribution which  any  such  cone  will  yield  to  the  surface  integral 
taken  over  T  of  a  mass  m  concentrated  at  0  is 


52  THE   NEWTONIAN   POTENTIAL  FUNCTION 

/  dSi  cos  ai    .    dS2  COS  02    ,    dSgCOStta   ,    dS4COSa4 

V      n"  ^2^  rg^  r/ 

=  m(  —  doi  +  dw  —  dw  +  cZo)  —  •••)  =  ??i.O  =  0, 

and  the  surface  integral  over  any  closed  surface  of  the  normal 
attraction  due  to  any  system  of  outside  masses  is  zero. 

The  results  proved  ahove  may  be  put  together  and  stated  in 
the  form  of  a 

Theorem  due  to  Gauss. 

If  there  he  any  distribution  of  matter  partly  within  and  partly 
without  a  dosed  surface  T,  and  if  31  he  the  sum  of  the  masses 
which  T  encloses^  and  M'  the  sum  of  the  masses  outside  T,  the 
surface  integral  over  T  of  the  normal  attraction  N  toward  the 
interior,  due  to  both  M  and  M',  is  equal  to  AttM.  If  V  be  the 
potential  function  due  to  both  31  and  31',  we  have 


Jy*=/. 


D  F-cZs  =  47r3f. 


It  is  easy  to  see  that  if  a  mass  31  be  supposed  concentrated 
on  the  surface  of  any  closed  surface  T  whose  curvature  is  every- 
where finite,  the  surface  integral  of  normal  attraction  taken 
over  Twill  be  27r3r',  for  all  the  elementary  cones  which  can  be 
drawn  from  a  point  P  in  the  surface  so  as  to  cut  Tonce  or  some 
other  odd  number  of  times  lie  on  one  side  of  the  tangent  plane 
at  the  point,  and  intercept  just  half  the  surface  of  the  sphere  of 
unit  radius  whose  centre  is  P. 

From  Gauss's  Theorem  it  follows  immediately  that  at  some 
parts  of  a  closed  surface  situated  in  a  field  of  force,  but  en- 
closing none  of  the  attracting  mass,  the  normal  component  of 
the  resultant  attraction  must  act  towards  the  interior  of  the 
surface  and  at  some  parts  toward  the  exterior,  for  otherwise 
the  limit  of  the  sum  of  the  intrinsically  positive  elements  of  the 
surface,  each  one  multiplied  by  the  component  in  the  direction 
of  the  interior  noraaal  of  the  attraction  at  one  of  its  own  points, 
could  not  be  zero.  In  other  words,  the  potential  function, 
whose  rate  of  change  measures  the  attraction,  must  at  some 


rN  THE  CASE   OF   GRAVITATION.  53 

parts  of  the  surface  increase  and  at  others  decrease  in  the  direc- 
tion of  the  interior  normal. 

From  this  it  follows  that  the  potential  function  cannot  have  a 
maximum  or  a  minimum  value  at  a  point  in  empty  space  ;  for 
if  al  such  a  point  Q  the  potential  function  had  a  maximum  value, 
we  could  surround  Q  by  a  small  closed  surface,  at  every  point 
of  which  the  potential  function  would  increase  in  the  direction 
of  the  interior  normal,  and  this  would  be  inconsistent  with  the 
fact  that  the  surface  integral  of  normal  attraction  taken  over 
the  surface,  which  would  contain  no  matter,  must  be  zero. 
Similarly  it  may  be  shown  that  the  potential  function  cannot 
have  a  minimum  value  at  a  point  in  empty  space. 

If  the  potential  function  be  constant  over  a  closed  surface 
which  contains  none  of  the  attracting  mass,  it  has  the  same 
value  throughout  the  interior ;  for  if  this  were  not  the  case, 
some  point  or  region  Q  within  T  would  have  a  value  greater  or 
less  than  the  surrounding  region,  and  we  could  enclose  Q.  bj-  a 
closed  surface  to  which  we  could  apply  the  course  of  reasoning 
just  used  to  show  that  V  cannot  attain  a  maximum  value  at  a 
point  in  empty  space. 

32.  Tubes  of  Force.  A  line  which  cuts  orthogonally  the  dif- 
ferent members  of  the  system  of  equipotential  surfaces  cor- 
responding to  any  distribution  of  matter  is  called  a  '-line  of 
force,"  since  its  direction  at  each  point  of  its  course  shows  the 
direction  of  the  resultant  force  at  the  point.  If  tlirough  all 
points  of  the  contour  of  a  portion  of  an  equipotential  surface 
lines  of  force  be  drawn,  these  lines  lie  on  a  surface   called  a 


Fig.  24. 


"tube  of  force."  We  may  easily  apply  Gauss's  Theorem  to  a 
space  cut  out  and  bounded  by  a  portion  of  a  tube  of  force  and 
two  equipotential  surfaces  ;  for  the  sides  of  the  tube  do  not  con- 


54  THE   NEWTONIAN   POTENTIAL  FUNCTION 

tribute  anything  to  the  surface  integral  of  normal  attraction,  and 
the  resultant  force  is  all  normal  at  points  in  the  equipotential 
surfaces.  If  w  and  w'  are  the  areas  of  the  sections  of  a  tube  of 
force  made  b}'  two  equipotential  surfaces,  and  if  F  and  F'  are 
the  average  interior  forces  on  w  and  w',  we  have 

i^(u+F'o)'  =  0  [87] 

if  the  tube  encloses  empt}"  space,  and 

Fa>+i^'o>'=47rW  [88] 

when  the  tube  encloses  a  mass  m  of  attracting  matter. 

33.  Spherical  Distributions.  In  the  case  of  a  distribution 
about  a  point  in  spherical  shells,  so  that  the  density  is  a 
function  of  the  distance  from  this  point  only,  the  lines  of  force 
are  straight  lines  whose  directions  all  pass  through  the  central 
point.  Every  tube  of  force  is  conical,  and  the  areas  cut  out  of 
different  equipotential  surfaces  by  a  given  tube  of  force  are  pro- 
portional to  the  square  of  the  distance  from  the  centre. 

Consider  a  tube  of  force  which  intercepts  an  area  i/^  from  a 
spherical  surface  of  unit  radius  drawn  with  0  as  a  centre,  and 
apply  Gauss's  Theorem  to  a  box  cut  out  of  this  tube  by  two 
equipotential  surfaces  of  radii  r  and  (?'  +  Ar)  respectively. 


Fig.  25. 

Let  AOB  (Fig.  25)  be  a  section  of  the  tube  in  question. 
The  area  of  the  portion  of  spherical  surface  w  which  is  repre- 
sented in  section  at  ad  is  r^i/',  and  the  area  of  that  at  he  is 
(r  +  ^rY\p.  If  the  average  force  acting  on  a>  toward  the  inside 
of  the  box  is  F,  the  average  force  acting  on  w'  toward  the  inside 
of  the  box  will  be  —  (7^+A,JP),  and  the  surface  integral  of 
normal  attraction  taken  all  over  the  outside  of  the  box  is 

Ft^^  -  {F+  ^,F)  (r  -f-  Ar)  V  =  -"A *  \{F'  r")  -       [89] 


IN  THE  CASE   OF   GRAVITATION.  55 

If  the  tube  of  force  which  we  have  been  considering  be  ex- 
tended far  enough,  it  will  cut  all  the  concentric  layers  of  matter, 
traverse  all  the  empty  regions  between  the  layers,  if  there  are 
such,  and  finally  emerge  into  outside  space. 

If  we  choose  r  so  that  the  box  shall  contain  no  matter,  the 
surface  integral  taken  over  the  box  must  be  zero. 

lu  this  case, 

therefore,  F=  -.  [90] 

and  r=-^_  +  yM.  [91] 

From  this  it  follows  that  in  a  region  of  empty  space,  either 
included  between  the  two  members  of  a  system  of  concentric 
spherical  shells  of  density  depending  only  upon  the  distance 
from  the  centre,  or  outside  the  whole  system,  the  force  of  attrac- 
tion at  different  points  varies  inversely  as  the  squares  of  the 
distances  of  these  points  from  the  centre. 

Suppose  that  the  box  (abed)  lies  in  a  shell  whose  densit}'  is 
constant ;  then  the  surface  integral  of  normal  attraction  taken 
over  the  box  is  equal  to  4:7  times  the  matter  within  the  box.  In 
this  case  the  quantity  of  matter  inside  the  box  is 

Pi^[ir  +  ^ry-r']±   or    p,//y-"A/- +  e, 

•iTT 

where  c  is  an  infinitesimal  of  an  order  higher  than  the  first. 

Therefore, 

-  ./.  A,  (Fr)  =  4  77  (  p  (/'J- Ar  +  c) , 

whence  F=-  i^^'  +  ^,  [1)2] 

3         r- 

and  F=  -  -  - 1  -pr^  +  h-  [93] 

r      o 


56  THE   NEWTONIAX   POTENTIAL   FUNCTION 

If  the  box  lies  iu  a  shell  whose  density  is  inversely  propor- 
tional to  the  distance  from  the  centre,  we  shall  have 


limit  M^)__^_/A' 


Ar=o— rr--^-4-(f)^.  [94] 


whence  F=-2TrX-{-^,  [95] 

and  V=---27rXr  +  a.  [901 

r 

In  general,  if  the  box  lies  in  a  shell  whose  density  is  /(r) ,  we 
shall  have 

iSo^^  =  -4-/W>^,  [97] 

whence  F=-^-~  Cf(r)  1^  •  dr.  [98] 

In  order  to  learn  how  to  use  the  results  just  obtained  to  de- 
termine the  force  of  attraction  at  any  point  due  to  a  given 
spherical  distribution,  let  us  consider  the  simple  case  of  a  single 
shell,  of  radii  4  and  5,  and  of  density  [Ar]  proportional  to  the 
distance  from  the  centre. 

At  points  within  the  cavity  enclosed  by  the  shell  we  must 
have,  according  to  [90]  and  [91], 

F=^,    and     F=--+ya; 
■f  r 

But  the  force  is  evidently  zero  at  the  centre  of  the  shell,  where 
r  is  zero,  so  that  c  must  be  zero  everywhere  within  the  cavity, 
and  there  is  no  resultant  force  at  any  point  in  the  region.  The 
value,  at  the  centre,  of  the  potential  function  due  to  the  shell  is 
evidently 

r\    .   o  J       2447r\  rof,-, 

/u,  =  I   4tTrX'rar= ,  [99] 

and  it  has  the  same  value  at  all  other  points  in  the  cavity. 

In  the  shell  itself  it  is  easy  to  see  that  we  must  have  at  all 
points, 

F=^-7rXr     and     F=---— +/x'.  [100] 


IX   THE   CASE   OF   GRAVITATION.  57 

In  order  to  determine  the  constants  in  this  equation,  we  may 
make  use  of  the  fact  that  F  and  V  are  everywhere  continuous 
functions  of  the  space  coordinates,  so  that  the  values  of  F  and 
V  obtained  by  putting  r  =  4,  the  inner  radius  of  the  shell,  in 
[100],  must  be  the  same  as  those  obtained  by  making  r  =  \  in 
the  expressions  which  give  the  values  of  F  and  V  for  the  cavity 
enclosed  by  the  shell.     This  gives  us 

c'  =  2567rX    and     /x'  =  ^^^^, 

so  that  for  points  within  the  mass  of  the  shell  we  have 

F=      IMz^_^Xr2,  [101] 

r- 

and  F= — +      „     •  [102] 

r  o  6 

For  points  without  the  shell  we  have  the  same  general  expres- 
sions for  F  and  V  as  for  points  within  the  cavity  enclosed  by 
the  shell,  namely, 

F=~,     and     V=--  +  m,  [10:3] 

1^  r 

but  the  constants  are  different  for  the  two  regions. 

Keeping  in  mind  the  fact  that  F  and  V  are  continuous,  it  is 
eas}'  to  see  that  we  must  get  the  same  result  at  the  boundary  of 
the  shell,  where  r=  5,  whether  we  use  [103],  or  [101]  and  [102]. 

This  gives 

^•  =  -3G9-A.     and     m  =  0; 

so  that  for  all  points  outside  the  shell  we  have 

p^_'}(^^^  [104] 

r 

and  F=      ^-^^^-  [105] 

r 

These  last  results  agree  with  the  statements  made  in  Section 
13,  for  the  mass  of  the  shell  is  300 -A. 

The  values,  at  every  point  in  space,  of  the  potential  function 
and  of  the  attraction  due  to  any  spherical  distribution  may  be 


58 


THE   NEWTONIAN   POTENTIAL   FUNCTION 


found  by  determining,  first,  with  the  aid  of  Gauss's  Theoreni, 
the  general  expressions  for  F  and  V  in  the  several  regions  ; 
then  the  constants  for  the  -innermost  region,  remembering  that 
there  is  no  resultant  attraction  at  the  centre  of  the  system ;  and 
finally,  in  succession  (moving  from  within  outwards),  the  con- 
stants for  the  otlier  regions,  from  a  consideration  of  the  fact 
that  no  abrupt  change  in  the  values  of  either  F  or  V  is  made  by 
crossing  the  common  boundary  of  two  regions. 

This  method  of  treating  problems  is  of  great  practical  im- 
portance. 

34.  Cylindrical  Distributions,  In  the  case  of  a  cylindrical 
distribution  about  an  axis,  where  the  density  is  a  function  of 
the  distance  from  the  axis  only,  the  equipotential  surfaces  are 
concentric  cylinders  of  revolution  ;  the  lines  of  force  are  straight 
lines  perpendicular  to  the  axis ;  and  every  tube  of  force  is  a 
wedge. 

If  we  apply  Gauss's  Theorem  to  a  box  shut  in  between  two 
equipotential  surfaces  of  radii  r  and  r  +  Ar,  two  planes  perpen- 
dicular to  the  axis,  and  two  planes  passing  through  the  axis. 


Fig.  26. 

we  have,  if  xp  is  the  area  of  the  piece  cut  out  of  the  cylindrical 
surface  of  unit  radius  by  our  tube  of  force, 

(o  =  r-{f/,     to'  =  (r  +  A?-)-i/^, 
and  for  the  surface  integral  of  normal  attraction  taken  over  the 
box. 


FiD  +  F'o>'  =  -xj/'A,(r'F). 

If  our  box  is  in  empty  space, 

A,(r-F)  =  Q, 


so  that 


F=       and     V  =  clogr -\- fi. 


[lOG] 


[107] 


IN  THE  CASE  OF   GRAVITATION.  59 

If  the  box  is  within  a  shell  of  constant  density  p, 
—  ij/  -  A,(r  •  F)  =  4  TTi/'  pr  Ar, 
so  that       F=j^  —  2irpr    and     V=c\ogr  —  Trpr^+fx.        [108] 

35.  Poisson's  Equation.  Let  us  now  apply  Gauss's  Theorem 
to  the  ease  where  our  closed  surface  is  that  of  an  element  of 
volume  of  an  attracting  mass  in  which  p  is  either  constant  or  a 
continuous  function  of  the  space  coordinates.  We  will  consider 
three  cases,  using  first  rectangular  coordinates,  then  cylinder 
coordinates,  and  finally  spherical  coordinates. 


B 

f^ 

^/ 

P, 

/ 

/ 

P~ 

A 

P^ 

N               X 

L 

v 

/ 

Fig.  27. 

I.  In  the  first  case,  our  element  is  a  rectangular  parallelopiped 
(Fig.  27).  Perpendicular  to  the  axis  of  x  are  two  equal  sur- 
faces of  area  Ay-A^,  one  at  a  distance  x  from  the  plane  yz^  and 
one  at  a  distance  x  +  Aa;  from  the  same  plane.  The  average 
force  perpendicular  to  a  plane  area  of  size  AyA2,  parallel  to  the 
plane  yz^  and  with  edges  parallel  to  the  axes  of  y  and  z,  is  evi- 
dently some  function  of  the  coordinates  of  the  corner  of  the 
element  nearest  the  origin. 

That  is,  if  P=  (.r,  ?/.  2),  the  average  force  on  PP^  parallel  to 
the  axis  of  x  is  X=/(.r,  ?/,  2),  and  the  average  force  on  r^P-  in 
the  same  direction  is  f{x  -f-  A.r,  //,  2)  =  X  -f-  A^  X,  so  that  PP^ 
and  PiP-  yield  towards  the  surface  integral  of  interior-normal 

attraction  taken  over  the  element,  the  quantitv  —  A.rAvAz-    '^  • 

A.r 

Similarly,    the   other  two   pairs  of   elementary  surfaces  yield 


60 


THE   NEWTONIAN   POTENTIAL   FUNCTION 


—  Aa;A?/Az^ —  and  —  Aa^AyAz^^ — ,  and,  if  po  is  the  average 

A?/  Az 

density  of  the  matter  enclosed  by  the  box,  we  have 
"A,X 


—  AxAyAz 


Ax. 


A,T_^A^Z' 


Ay 


Az 


A-n-p^AxAyAz.    [109] 


This  equation  is  true  whatever  the  size  of  the  element  Ao;  Ay  As:. 
If  this  element  is  made  smaller  and  smaller,  the  average  nor- 
mal force  [X]  on  PP4  approaches  in  value  the  force  l^D^V]  at 
P  in  the  direction  of  the  axis  of  a; ;  Y"  and  Z  approach  respec- 
tively the  limits  DyVsLndD^V;  and  p^  approaches  as  its  limit 
the  actual  density  [p]  at  P. 

Taking  the  limits  of  both  sides  of  [109],  after  dividing  by 
AxAyAz,  we  have 

or  v'F=-47rp,  [110] 

which  is  Poisson's  Equation.  The  potential  function  due  to  any 
conceivable  distribution  of  attracting  matter  must  be  such  that 
at  all  points  within  the  attracting  mass  this  equation  shall  be 
satisfied. 

For  points  in  empty  space  p  =  0,  and  Poisson's  Equation 
degenerates  to  Laplace's  Equation. 

II.  In  the  case  of  cylindrical  coordinates,  the  element  of  vol- 
ume (Fig.  28)  is  bounded  by  two  cylindrical  surfaces  of  revo- 


FiG.  28. 

lution  having  the  axis  of  z  as  their  common  axis  and  radii  r  and 
r  +  A?-,  two  planes  perpendicular  to  this  axis  and  distant  Az 


IN   THE  CASE   OF   GRAVITATION. 


61 


from  each  other,  and  two  planes  passing  through  the  axis  and 
forming  with  each  other  the  diedral  angle  A^. 

Call  R,  0,  and  Z  the  average  normal  forces  upon  the  elemen- 
tary planes  PPci  PP21  and  PP3  respectively,  then  the  surface 
integral  of  normal  attraction  over  the  volume  element  will  be 

-  AeAzX{r-E)  —  Ar  A2  Ag©  -  A^  [rAr  +  |-(Ar)=']A^Z 

=  47rp,  (vol.  ofbox)  ;  [111] 

whence,  approximately, 


1  A,(rJg)      1  A,Q 
Ar  r   Ae 


r 


A,Z  .        vol.  of  box     ri  1  .on 

=  -^^Po       .     .^. —     [112] 


Az 


rArAOAz 


The  force  at  Pin  direction  PPr,  is  D^F,  in  direction  PP4  is  D^  V, 
and  perpendicular  to  LP  in  the  plane  PLP^  is  -  •  1)^  F,  so  that 
if  the  box  is  made  smaller  and  smaller,  our  equation  approaches 
the  form      1^^(,.. j^^^^.^  1  £,^2^_^^^2^^_4^p.  ["113] 

T  IT 


Fig.  29. 


III.    In  the  case  of  spherical  coordinates,  the  volume  element 
is  of  the  shape  shown  in  Fig.  29.     Let  OP=r,  ZOP=e,  and 


62  THE  NEWTONIAN   POTENTIAL  FUNCTION 

denote  by  (f>  the  diedral  angle  between  the  planes  ZOP  and 
ZOX.  Denote  by  M,  ©,  and  ^  the  average  normal  forces  on  the 
faces  PPg,  PP^-,  and  PP2  respectively  ;  then  the  surface  integral 
of  normal  attraction  over  the  elementary  box  is  approximately 

—  sin^A^A*^  •  A,('rB)  —  r  A(9Ar  A^$  —  r  A<^  Ar  •  Ag(sin(9  •  0) 

=  47r/3o-(vol.  of  box)  ;  [H^^] 

whence   J_.  Mr^J?) +_!_.  A^ +  _L_.  A.(sin^.Q) 
r^        Ar  rsin^    A<^       rsin^  A6 

.  vol.  of  box  ri  1  --I 

'^'*  ?-2sin^ArA^A</)  *-       -^ 

The  force  at  P  in  the  direction  PP5  is  Z),F,  in  the  direction 

PPi  is  — ; Drf,  V,  and  in  the  direction  PP4  is  --DgV;  there- 

r  sm  0  r 

fore,  as  the  element  of  volume  is  made  smaller  and  smaller,  our 

equation  approaches  the  form 

sin  6 
=  -4:7rpr'sme.  [IIG] 

This  equation,  as  well  as  that  for  cylinder  coordinates,  might 
have  been  obtained  by  transformation  from  the  equation  in 
rectangular  coordinates. 

36.  Poisson's  Equation  in  the  Integral  Form.  In  [109]  X 
may  be  regarded  as  a  function  of  x,  ?/,  z,  Ay,  and  Az,  which  ap- 
proaches Z>^F'as  a  limit  when  Ay  and  Az  are  made  to  approacli 
zero,  and  it  may  not  be  evident  that  the  limit,  when  Ax,  Ay,  and 

Az  are  together  made  to  approach  zero,  of  the  fraction  -^-^  is 

D^V.  For  this  reason  it  is  worth  while  to  establish  Poisson's 
Equation  by  another  method. 

It  is  shown  in  Section  29  that  the  volume   integral  of  the 

quantity  —DA-\,  taken  throughout  a  certain  region,  is  the  sur- 


IN   THE  CASE   OF   GRAVITATION.  63 

face  integral  of  ^eosa  taken  all  over  the  surface  which  bounds 
r 

the  region.     In  this  proof  we  might  substitute  for  -  any  other 

function  of  the  three  space  coordinates  which  tlu-oughont  the 
region  is  finite,  continuous,  and  single- valued,  and  state  the 
results  iu  the  shape  of  the  following  theorem : 

If  T  is  an}'  closed  surface  and  U  a  function  of  x,  y,  and  z 
which  for  every  point  inside  T  has  a  finite,  definite  value  wliich 
changes  continuously  in  moving  to  a  neighboring  point,  then 

I    I    i  D^U '  clx  dy  dz  =  —  I  U  cos  ads,  [H"] 

r  r  Cd^  U-  dx  dydz=-  Cu cos  /Sds,  [11 8] 

and  r  r  Cd,  U'  dx  dy  dz  =  -  Cu  cos  yds,  [119] 

where  a,  /S,  and  y  are  the  angles  made  by  the  interior  normals 
at  the  various  points  of  the  surface  with  the  positive  direction 
of  the  coordinate  axes,  and  wliere  the  sinister  integrals  are  to  be 
extended  all  through  the  space  enclosed  by  T,  and  the  de:xter 
integrals  all  over  the  bounding  surface. 

If  we  apply  this  theorem  to  an  imaginary  closed  surface  which 
shuts  in  any  attracting  mass  of  density  either  uniform  or  vari- 
able, and  if  for  t"  in  [11  7],  [UH],  and  [119]  we  use  resi)ectively 
D^V,  D^V,  and  D^V,  and  adil  tlie  resulting  equations  together, 
we  shall  have 

fff(I^K'V+  DiV+  I);V)dxdydz 

=  -C{D,  Vcosa  +  D,  Vcos(3  +  D,  V  cosy)  ds.       [  1 20] 

The  integral  in  the  second  member  of  this  equation  is  evi- 
dently (see  ['')<>])  the  surface  integral  of  normal  attraction  taken 
over  our  imaginary  closed  surface,  and  this  by  (iauss's  Theorem 
is  equal  to  -iTr  times  the  quantity  of  matter  inside  the  surface, 
so  that 


G4  THE  NEWTONIAN   POTENTIAL   FUNCTION 

=  —  477  I   I    \  pdxdydz.  [121] 

Since  this  equation  is  true  whatever  the  form  of  the  closed 
surface,  we  must  have  at  every  point 

D,2F+ Z>/F+ Z>/F=  -  47rp. 

For  if  throughout  any  region  V  Fwere  greater  than  —  47rp,  we 
miglit  take  the  boundary  of  this  region  as  our  imaginary  surface. 
In  this  case  every  term  in  tlie  sum  wliose  limit  gives  the  sinister 
of  [121]  would  be  greater  than  the  corresponding  term  in  the 
dexter,  so  that  the  equation  would  not  be  true.  Similar  reason- 
ing shuts  out  the  possibility  of  V"F's  being  less  than  —  47rp. 

37.  The  Average  Value  of  the  Potential  Function  on  a  Spheri- 
cal Surface.  If,  in  a  field  of  force  due  to  a  mass  m  concentrated 
at  a  point  P,  we  imagine  a  spherical  surface  to  be  drawn  so  as 
to  exclude  P,  the  surface  integral  taken  over  this  surface  of  the 
value  of  the  potential  function  due  to  m  is  equal  to  the  area  of 
the  surface  multiplied  by  the  value  of  the  potential  function  at 
the  centre  of  the  sphere. 

To  prove  this,  let  the  radius  of  the  sphere  be  a  and  the  dis- 
tance [_0P~\  of  P  from  its  centre  c.  Take  the  centre  of  the 
sphere  for  origin  and  the  line  OP  for  the  axis  of  x.  Divide  the 
surface  of  the  sphere  into  zones  by  means  of  a  series  of  planes 
cutting  the  axis  of  x  perpendicularly  at  intervals  of  Ax*.  The 
area  of  each  one  of  these  zones  is  2Tradx,  so  that  the  surface 

integral  of  —  is 

/+"      m'JTradx       _  _  [2  -n-m  a-\/o?  +  c?  —  2  ex 
»    ■\/a?  +  (?-2cx         L       •  c 

and  the  value  of  this,  since  the  radical  represents  a  positive 

quantity,  is  — ,  which  proves  the  proposition. 

c 


IX  THE   CASE   OF   GRAVITATION.  65 

The  surface  integral  of  the  potential  function  taken  over  the 
sphere  divided  by  the  area  of  the  sphere  is  often  called  "the 
average  value  of  the  potential  function  on  the  spherical  sur- 
face." 

If  we  have  any  distribution  of  attracting  matter,  we  may 
divide  it  into  elements,  apply  the  theorem  just  proved  to  each 
of  these  elements,  and,  since  the  potential  function  due  to  the 
whole  distribution  is  the  sum  of  those  due  to  its  parts,  assert 
that : 

The  average  value  on  a  spherical  surface  of  the  potential 
function  due  to  any  distribution  of  matter  entirely  outside  the 
sphere  is  equal  to  the  value  of  the  potential  function  at  the 
centre  of  tlie  sphere. 

It  follows,  from  this  theorem,  that  if  the  potential  function  is 
constant  within  any  closed  surface  S  drawn  in  a  region  T,  which 
contains  no  matter,  so  as  to  shut  in  a  part  of  that  region,  it  will 
have  the  same  value  in  those  parts  of  T  which  lie  outside  aS". 
For,  if  the  values  of  the  potential  function  at  points  in  empty 
space  just  outside  S  were  different  from  the  value  inside,  it  would 
always  be  possible  to  draw  a  sphere  enclosing  no  matter  whose 
centre  should  be  inside  S,  and  which  outside  *S'  should  include 
only  such  points  as  were  all  at  either  higher  or  lower  potential 
than  the  space  inside  »S' ;  but  in  this  case  the  value  of  the  poten- 
tial function  at  the  centre  of  the  sphere  would  not  be  the  average 
of  its  values  over  its  surface. 

The  value  of  the  potential  function  cannot  be  constant  in  im- 
limited  empty  space  surrounding  an  attracting  mass  3/,  for,  if 
it  were,  we  could  suiTOund  the  mass  by  a  surface  over  which  the 
surface  integral  of  normal  attraction  would  be  zero  instead  of 
47r3/. 

The  average  value  on  a  spherical  surface  of  the  potential 
function  [F],  due  to  any  distribution  [3/]  of  attracting  matter 
wholly  within  the  surface,  is  the  same  as  if  M  were  concen- 
trated at  the  centre  0  of  the  space  which  the  surface  encloses. 
For  the  average  values  [T",  and  Kj  +  A^T',]  of  V  on  con- 
centric spherical  surfaces  of  radii  r  and  /'  -f  Ar  may  be  written 


66  THE  NEWTONIAN  POTENTIAL  FUNCTION 

j  Vds  (or   —  I  Vdu),  if  dw  is  tlie  solid  angle  of  an  ele- 

mcntary  cone  with  vertex  at  O,  which  intercepts  the  element  ds 

from  the  surface  of  a  sphere  of  radius  7'),  and  —  (  (F+A,F)d(o  ; 

47r»/ 

whence  A^F^  =  —  JArF-doj, 

4:TrJ 

and  D,Vo  =  —  CD,V-do>. 

4:7rJ 

Now  —  i  D^V-  a?d(a  is  the  integral  of  normal  attraction  taken 
over  the  spherical  surface,  whence,  by  Gauss's  Theorem, 

D,Vq  =  ---—,       and       Fo  =  — +0, 
47rr  r 

since  "F)  =  0,  for  r=oo. 

38.  The  Equilibrium  of  Fluids  at  Rest  under  the  Action  of 
Given  Forces.  Elementary  principles  of  Hydrostatics  teach  us 
tliat  when  an  incompressible  fluid  is  at  rest  under  the  action  of 
any  system  of  applied  forces,  the  hydrostatic  pressure  p  at  the 
point  (a;,  y,  z)  must  satisfy  the  differential  equation 

dp  =  p{Xdx-\-Ydy-\-Zdz),  [122] 

where  X,  Y",  and  Z  are  the  values  at  that  point  of  the  force 
applied  per  unit  of  mass  to  urge  the  liquid  in  directions  parallel 
to  the  coordinate  axes. 

For,  if  we  consider  an  element  of  the  liquid  [Ao^AyA^] 
(Fig.  27)  whose  average  density  is  p^  and  whose  corner  next 
the  origin  has  the  coordinates  (a?,  y,  z) ,  and  if  we  denote  by  p^ 
the  average  pressure  per  unit  surface  on  the  face  FP^PiPs,  by 
p^  +  A^p^  the  average  pressure  on  the  face  PiPuP^Pf,^  and  by 
Xn  the  average  applied  force  per  unit  of  mass  which  tends  to 
move  the  element  in  a  direction  parallel  to  the  axis  of  x,  we 
have,  since  the  element  is  at  rest, 

p^  A?/  A2!  +  pu  Xo  Ax  AyAz==( i\  -\-  A^p,)  Ay  Az, 

\r  A^p^ 

or  po-X.o  =  -f^. 

Ax 


IN   THE  CASE  OF   GRAVITATION.  67 

If  the  element  be  made  smaller  and  smaller,  the  first  side  of 
the  equation  approaches  the  limit  pX,  and  the  second  side  the 
limit  D^p,  where  p  is  the  hydrostatic  pressure,  equal  in  all  direc- 
tions, at  the  point  P. 

This  gives  us  D^p  =  pX.  [1 23] 

In  a  similar  manner,  we  may  prove  that 
D^p  =  pY, 
and  D^p  =  pZ; 

whence  dp  =  D^p  dx  +  D^p  dy  -\-  D^pdz 

=  p{Xdx  +  Ydy  +  Zdz) . 

•  If  in  any  case  of  a  liquid  at  rest  the  only  external  force 
applied  to  each  particle  is  the  attraction  due  to  some  outside 
mass,  or  to  the  other  particles  of  the  liquid,  or  to  both  together, 
X,  Y,  and  Z  are  the  partial  derivatives  with  regard  to  x,  y,  and 
2  of  a  single  function  F,  and  we  may  write  our  general  equation 
in  the  form 

dp  =  p{D,V-  dx  +  D,V'  dy  +  D,V-  dz)  =  p'dV, 
whence,  if  p  is  constant, 

p  =  pF+ const.,  [124] 

and  the  surfaces  of  equal  hydrostatic  pressiu-e  are  also  equi- 
potential  surfaces. 

According  to  this,  the  free  bounding  surfaces  of  a  liquid  at 
rest  under  the  action  of  gravitation  only  are  equipotential. 

EXAMPLES. 

1.  Prove  that  a  particle  cannot  be  in  stable  equilibrium  under 
the  attraction  of  any  system  of  masses.     [Earnshaw.] 

2.  Prove  that  if  all  the  attracting  mass  lies  without  an  equi- 
potential surface  S  on  which  V=  c,  then  F=  c  in  all  space 
inside  S. 

3.  Prove  that  if  all  the  attracting  mass  lies  within  an  equi- 
potential surface  S  on  which  V=C,  then  in  all  space  outside  S 
the  value  of  the  potential  function  lies  between  C  and  0. 


68  THE  NEWTONIAN  POTENTIAL  FUNCTION 

4.  The  source  of  the  Mississippi  River  is  nearer  the  centre  of 
the  earth  than  the  mouth  is.  What  can  be  inferred  from  this 
about  the  slope  of  level  surfaces  on  the  earth? 

5.  If  in  [59]  X  be  made  equal  to  zero,  V  becomes  infinite. 
How  can  j^ou  reconcile  this  with  what  is  said  in  the  first  part  of 
Section  22? 

6.  Are  all  solutions  of  Laplace's  Equation  possible  values  of 
the  potential  function  in  empty  space  due  to  distributions  of 
matter  ?  Assume  some  particular  solution  of  this  equation 
which  will  serve  as  the  potential  function  due  to  a  possible  dis- 
tribution and  show  what  this  distribution  is. 

7.  If  the  lines  of  force  which  traverse  a  certain  region  are 
parallel,  what  may  be  inferred  about  the  intensity  of  the  force 
within  the  region  ? 

8.  The  path  of  a  material  particle  starting  from  rest  at  a 
point  P  and  moving  under  the  action  of  the  attraction  of  a  given 
mass  3/ is  not  in  general  the  line  of  force  due  to  Jf  which  passes 
through  P.  Discuss  this  statement,  and  consider  separately 
cases  where  the  lines  of  force  are  straight  and  where  they  are 
curved. 

9.  Draw  a  figure  corresponding  to  Figure  17  for  the  case  of 
a  uniform  sphere  of  unit  radius  surrounded  by  a  concentric 
spherical  shell  of  radii  2  and  3  respectively. 

10.  Draw  with  the  aid  of  compasses  traces  of  four  of  the 
equipotential  surfaces  due  to.  two  homogeneous  infinite  cylinders 
of  equal  density  whose  axes  are  parallel  and  at  a  distance  of 
5  inches  apart,  assuming  the  radius  of  one  of  the  cylinders  to 
be  1  inch  and  that  of  the  other  to  be  2  inches. 

11.  Draw  with  the  aid  of  compasses  meridian  sections  of 
four  of  the  equipotential  surfaces  due  to  two  small  homogeneous 
spheres  of  mass  m  and  2  m  respectively,  whose  centres  are  4 
inches  apart.  Can  equipotential  surfaces  be  drawn  so  as  to  lie 
wholly  or  partly  within  one  of  the  spheres?  What  value  of  the 
potential  function  gives  an  equipotential  surface  shaped  like 
the  figure  8  ?  Show  that  the  value  of  the  resultant  force  at  the 
point  where  this  curve  crosses  itself  is  zero. 


IN  THE  CASE  OF  GRAVITATION.  69 

12.  A  sphere  of  radius  3  inches  and  of  constant  density  fx  is 
surrounded  by  a  spherical  sliell  concentric  with  it  of  radii  4 
inches  and  5  inches  and  of  density  fir,  where  r  is  the  distance 
from  the  centre.  Compute  the  values  of  the  attraction  and  of 
the  potential  function  at  all  points  in  space  and  draw  curves  to 
illustrate  the  fact  that  V  and  Z),F  are  everywhere  continuous 
and  that  DJ^ Vis  discontinuous  at  certain  points. 

13.  A  very  long  cylinder  of  radius  4  inches  and  of  constant 
density  /a  is  surrounded  by  a  cylindrical  shell  coaxial  with  it 
and  of  radii  6  inches  and  8  inchefe.  The  density  of  this  shell  is 
inversely  proportional  to  the  square  of  the  distance  from  the 
axis,  and  at  a  point  8  inches  from  this  axis  is  fx.  Use  the  Theo- 
rem of  Gauss  to  find  the  values  of  F,  D^V,  and  D,^V  at  differ- 
ent points  on  a  line  perpendicular  to  the  axis  of  the  cj'linder  at 
its  middle  point.  If  the  value  of  the  attraction  at  a  distance 
of  20  inches  from  the  axis  is  10,  show  how  to  find  /*. 

14.  Use  Dirichlet's  value  of  Z>^F,  given  by  equation  [78], 
to  find  the  attraction  in  the  direction  of  the  axis  of  x  at  points 
within  a  spherical  shell  of  radii  r^  and  rj  and  of  constant  den- 
sity p. 

15.  Are  there  any  other  cases  except  those  in  which  the 
density  of  the  attracting  matter  depends  only  upon  the  distance 
from  a  plane,  from  an  axis,  or  from  a  central  point,  where  sur- 
faces of  equal  force  are  also  equipotential  surfaces?  Prove 
your  assertion. 

16.  Prove  that  if  a  mass  Mi  be  divided  up  into  elements,  and 
if  each  one  of  these  elements  be  multiplied  by  the  value  at  one 
of  its  own  points  of  the  potential  function  Vo  due  to  another 
mass  Mo,  the  limit  of  the  sum  of  these  infinitesimal  products  will 
be  equal  to  the  limit  of  the  sum  extended  over  Mo  of  the  product 
of  the  masses  of  its  elements  by  the  corresponding  values  of  the 
jjotential  function  due  to  Mi.     That  is,  show  that 

fv,-(lMi=  CVi-dMo, 

where  the  sinister  integral  is  to  embrace  all  Mi  and  the  dexter 
all  Mi.     [Gauss.] 


mi  m<. 
l- 


y=yi    r-Tt«-     ,  •     -, 
[Minchin.] 

y=y\ 


70  THE   NEWTONIAN    POTENTIAL  FUNCTION 

17.  Two  uniform  straight  wires  of  length  I  and  of  masses  mi 
and  m2  are  parallel  to  each  other  and  perpendicular  to  the  line 
joining  their  middle  points,  which  is  of  length  ?/i.  Show  that 
the  amount  of  work  required  to  increase  the  distance  between 
the  wires  to  2/2  by  moving  one  of  them  parallel  to  itself  is 

y_VFT?-nog^^'  +  ^'~^ 
y 

18.  Show  that  if  the  earth  be  supposed  spherical  and  covered 
with  an  ocean  of  small  depth,  and  if  the  attraction  of  the  par- 
ticles of  water  on  each  other  be  neglected,  the  ellipticity  of  the 
ocean  spheroid  will  be  given  by  the  equation, 

The  centrifugal  force  at  the  equator 

-  -  ■• 

19.  A  spherical  shell  whose  inner  radius  is  r  contains  a  mass 
m  of  gas  which  obeys  the  Law  of  Boyle  and  Mariotte.  Find 
the  law  of  density  of  the  gas,  the  total  normal  pressure  on  the 
inside  of  the  containing  vessel,  and  the  pressure  at  the  centre. 

20.  If  the  earth  were  melted  into  a  sphere  of  homogeneous 
liquid,  what  would  be  the  pressure  at  the  centre  in  tons  per 
square  foot  ?  If  this  molten  sphere  instead  of  being  homo- 
geneous had  a  surface  density  of  2.4  and  an  average  density  of 
5.G,  what  would  be  the  pressure  at  the  centre  on  the  supposition 
that  the  density  increased  proportionately  to  the  depth? 

21.  A  solid  sphere  of  attracting  matter  of  mass  m  and  of 
radius  r  is  surrounded  by  a  given  mass  M  of  gas  which  obeys 
the  Law  of  Boyle  and  Mariotte.  If  the  whole  is  removed  from 
the  attraction  of  all  other  matter,  find  the  law  of  density  of  the 
gas  and  the  pressure  on  the  outside  of  the  sphere. 

22.  The  potential  function  within  a  closed  surface  S  due  to 
matter  wholly  outside  the  surface  has  for  extreme  values  the 
extreme  values  upon  S. 

23.  If  the  potential  functions  V  and  F'  due  to  two  systems 
of  matter  without  a  closed  surface  have  the  same  values  at  all 
points  on  the  surface,  they  will  be  equal  throughout  the  space 
enclosed  by  the  surface. 


IK  THE  CASE  OF   GRAVITATION.  71 

24.  The  potential  function  outside  of  a  closed  surface  due  to 
matter  wholly  within  the  surface  has  for  its  extreme  values  two 
of  the  following  three  quantities :  zero  and  the  extreme  values 
upon  the  surface. 

25.  Prove  that  if  R  is  the  distance  from  the  origin  of  coordi- 
nates to  the  point  P,  and  if  Vp  is  the  value  at  P  of  the  potential 
function  of  any  system  of  attracting  masses  within  a  finite  dis- 
tance of  the  origin,  the  limit  as  R  is  made  infinite  of  Vp-P  is 
equal  to  M,  the  whole  quantity  of  attracting  matter. 


72  THE  POTENTIAL  FUNCTION 


CHAPTER   HI. 

TEE  POTENTIAL  PUNOTION  IN  THE  CASE  OP 
EEPULSION. 

39.  Repulsion,  according  to  the  Law  of  Nature.  Certain 
physical  phenomena  teach  us  that  bodies  ma}'  acquire,  by 
electrification  or  otherwise,  the  property  of  repelling  each  other, 
and  that  the  resulting  force  of  repulsion  between  two  bodies  is 
often  much  greater  than  the  force  of  attraction  which,  ac- 
cording to  the  Law  of  Gravitation,  every  body  has  for  every 
other  body. 

Experiment  shows  that  almost  every  such  case  of  repulsion, 
however  it  may  be  explained  physicall}',  can  be  quantitatively 
accounted  for  by  assuming  the  existence  of  some  distribution  of 
a  kind  of"  matter,"  every  particle  of  which  is  supposed  to  repel 
every  other  particle  of  the  same  sort  according  to  the  "  Law  of 
Nature,"  that  is,  roughly  stated,  with  a  force  directly  propor- 
tional to  the  product  of  the  quantities  of  matter  in  the  particles, 
and  inversely  proportional  to  the  square  of  the  distance  between 
their  centres. 

In  this  chapter  we  shall  assume,  for  the  sake  of  argument, 
that  such  matter  exists,  and  proceed  to  discuss  the  effects  of 
different  distributions  of  it.  Since  the  law  of  repulsion  which 
we  have  assumed  is,  witii  the  exception  of  the  opposite  direc- 
tions of  the  forces,  mathematically  identical  with  the  law  which 
governs  the  attraction  of  gravitation  between  particles  of  pon- 
derable matter,  we  shall  find  that,  by  the  occasional  intro- 
duction of  a  change  of  sign,  all  the  formulas  which  we  have 
proved  to  be  true  for  cases  of  attraction  due  to  gravitation 
can  be  made  useful  in  treating  corresponding  problems  in 
repulsion. 


IN  THE  CASE  OF   REPULSION. 


73 


40.  Force  at  Any  Point  due  to  a  Given  Distribution  of 
Repelling  Matter.  Two  equal  quantities  of  repelling  matter 
concentrated  at  points  at  the  unit  distance  apart  are  called 
"  unit  quantities"  when  they  are  such  as  to  make  the  force  of 
repulsion  between  them  the  unit  force. 

If  the  ratio  of  the  quantity  of  repelling  matter  within  a  small 
closed  surface  supposed  drawn  about  a  point  P,  to  the  volume 
of  the  space  enclosed  by  the  surface,  approaches  the  limit  p  when 
the  surface  (always  enclosing  P)  is  supposed  to  be  made  smaller 
and  smaller,  p  is  called  the  "density"  of  the  repelling  matter 
at  P. 

In  order  to  find  the  magnitude  at  an}-  point  P  of  the  force  due 
to  an\'  given  distribution  of  repelling  matter,  we  may  suppose 
the  space  occupied  by  this  matter  to  be  divided  up  into  small 
elements,  and  compute  an  approximate  value  of  this  force  on  the 
assumption  that  each  element  repels  a  unit  quantity  of  matter 
concentrated  at  P  with  a  force  equal  to  the  quantity  of  matter 
in  the  element  divided  by  the  square  of  the  distance  between  P 
and  one  of  the  points  of  the  element.  The  limit  approached  by 
this  approximate  value  as  the  size  of  the  elements  is  diminished 
indefinitely  is  the  value  required. 


Fig.  30. 


Let  Q  (Fig.  30),  whose  coordinates  are  x',  ?/',  z\  be  the 
corner  next  the  origin  of  an  element  of  the  distribution.  Let'p 
be  the  density  at  Q  and  A.i-'Ay'A2'  the  volume  of  the  element ; 
then  the  force  at  P  due  to  the  matter  in  the  element  is  ai)proxi- 


74  THE  POTENTIAL  FUNCTION 

D  A^  \fj  \y. 

mately  equivalent  to  a  force  of  magnitude  f— — — ^ acting  in 

the  direction  QP,  or  a  force  of  magnitude  —  ^ — — ^ acting 

in  tlie  direction  PQ.  If  tlie  coordinates  of  P  are  cc,  y,  z,  the 
component  of  this  force  in  the  direction  of  the  positive  axis  of  x 

IS  — — ; f- — — f -^^ — ,  ,  ^    ,,-,,,  and  the  force  at  P  parallel 

to  the  axis  of  x  due  to  the  whole  distribution  of  repelling 
matter  is 

X=-CC  C              pix'-x)dx'dy'dz'  r.         -, 

JJJl{x'-xy--\-{y'-yy-\-{z'-zr]e  L       aJ 

where  the  triple  integration  is  to  be  extended  over  the  whole 
space  filled  with  the  repelling  matter.  For  the  components  of 
the  force  at  P  parallel  to  the  other  axes  we  have,  similarly, 


r=-CC  C  p(y'-y)dx'dy'dz'  . 

J  J  J  l(^x'-xy+{y'-yy+iz'-zyy  L       Bj 
and 

7-^      CCC  p(z'-z)dx'dy'dz'  pjg.  I 

JJJ[{x'-xy+  (y'-  yy+  (z'-  zyji  •-     ^^ 

If  we  denote  by  V  the  function  * 


/// 


pdx'dy'dz' 


l(x'-xy+(y'-yy+{z'-zy:\i 


2Ti' 


[126] 


which,  together  with  its  first  derivatives,  is  everywhere  finite 
and  continuous,  as  we  have  shown  in  the  last  chapter,  it  is  easy 
to  see  that 

X=-AF,     Y=-D,V,     Z  =  -D,V,  [127] 


E=y/{D,vy+iv,vy+{n,vy,  [i28] 

and  that  the  direction-cosines  of  the  line  of  action  of  the  re- 
sultant force  at  P  are 


IN  THE  CASE  OF   EEPULSION.  75 

It  follows  from  this  (see  Section  21)  that  the  component  in 
any  direction  of  the  force  at  a  point  P  due  to  any  distribution 
M  of  repelling  matter  is  minus  the  value  at  P  of  the  partial 
derivative  of  the  function  F  taken  in  that  direction. 

The  function  Fgoes  by  the  name  of  the  Newtonian  potential 
function  whether  we  are  dealing  with  attracting  or  repelling 
matter. 

In  the  case  of  repelling  matter,  it  is  evident  that  the  resultant 
force  on  a  particle  of  the  matter  at  any  point  tends  to  drive  that 
particle  in  a  direction  which  leads  to  points  at  which  the  poten- 
tial function  has  a  lower  value,  whereas  in  the  case  of  gravita- 
tion a  particle  of  ponderable  matter  at  any  point  tends  to  move 
in  a  direction  along  which  the  potential  function  increases. 

41.  The  Potential  Function  as  a  Measure  of  Work.  It  is 
easy  to  show  by  a  metliod  like  that  of  Article  27  that  the 
amount  of  work  required  to  move  a  unit  quantity  of  repelling 
matter,  concentrated  at  a  point,  from  Pj  to  P.,,  in  face  of  the 
force  due  to  any  distribution  M  of  the  same  kind  of  matter,  is 
Fa  —  Fi,  where  Vi  and  V2  Jire  the  values  at  P^  and  P.,  respec- 
tively of  the  potential  function  due  to  M.  The  farther  P,  is 
from  the  given  distribution,  the  smaller  is  Fi,  and  the  less  does 
V2  —  Fi  differ  from  Fo.  In  fact,  the  value  of  the  potential 
function  at  the  point  Po,  wherever  it  may  be,  measures  the  work 
which  would  be  required  to  move  the  unit  quantity  of  matter  by 
any  path  from  ''  infinity"  to  Pj. 

42.  Gauss's  Theorem  in  the  Case  of  Repelling  Matter.  If  a 
quantity  m  of  repelling  matter  is  concentrated  at  a  point  within 
a  closed  oval  surface,  the  resultant  force  due  to  m  at  any  point 
on  the  surface  acts  toward  the  outside  of  the  surface  instead  of 
towards  the  inside,  as  in  the  case  of  attracting  matter. 

Keeping  this  in  mind,  we  may  repeat  the  reasoning  of  Article 
31,  using  repelling  matter  instead  of  attracting  matter,  and  sub- 
stituting all  through  the  work  the  exterior  normal  for  the  in- 
terior normal,  and  in  this  way  prove  that : 


76  THE  POTENTIAL  FUNCTION 

If  there  be  any  distribution  of  repelling  matter  partly  within 
and  partly  without  a  closed  surface  T,  and  if  M  be  the  whole 
quantity  of  this  matter  enclosed  by  2',  and  M'  the  quantity  out- 
side T,  the  surface  integral  over  T  of  the  component  in  the  di- 
rection of  the  exterior  normal  of  the  force  due  to  both  M  and  M' 
is  equal  to  4  ttM.  If  V  be  the  potential  function  due  to  M  and 
Jf',  we  have  ^D^V-ds^^^M. 

43.  Poisson's  Equation  in  the  Case  of  Repelling  Matter.     If 

we  apply  the  theorem  of  the  last  article  to  the  surface  of  a 
volume  element  cut  out  of  space  containing  repelling  matter, 
and  use  the  notation  of  Article  35,  we  shall  find  that  in  the  case 
of  rectangular  coordinates  the  surface  integral,  taken  over  the 
element,  of  the  component  in  the  direction  of  the  exterior 
normal  is 


AxA?/A2; 


A.X    ,    A.,r  ,   A,Z" 


=  47rpo.Aa;A?/A2,      [130] 


Ax  A?/  Az 

where  X  is  the  average  component  in  the  positive  direction  of 
the  axis  of  x  of  the  force  on  the  elementary  surface  A?/A2!,  and 
where  Y  and  Z  have  similar  meanings.  It  is  evident  that  if 
the  element  be  made  smaller  and  smaller,  X,  Y,  and  Z  will 
approach  as  limits  the  components  parallel  to  the  coordinate 
axes  of  the  force  at  P.  These  components  are  —D^V,  —D^V, 
and  —D^V',  so  that  if  we  divide  [130]  by  Ax^y^z  and  then 
decrease  indefinitely  the  dimensions  of  the  element,  we  shall 
arrive  at  the  equation 

V'F=-47r/3.  [131] 

By  using  successively  cylinder  coordinates  and  spherical  co- 
ordinates we  may  prove  the  equations 

^A(ri),F)  +  ii)/F+ Z)/F=-47rp,  [132] 

and  sme-Dr{)^DrV)-\-^^^+Dg(sin6-DgV) 

=  -47rpr2sin^,  [133] 


IN  THE  CASE   OF   REPULSION.  77 

80  that  Poisson's  Equation  holds  whether  we  are  dealing  with 
attracting  or  repelling  matter. 

44.  Coexistence  of  Two  Kinds  of  Active  Matter.  Certain 
physical  phenomena  may  be  most  conveniently  treated  mathe- 
matically by  assuming  the  coexistence  of  two  kinds  of  "mattei*" 
such  that  any  quantity  of  either  kind  rei)els  all  other  matter  of 
the  same  kind  according  to  the  Law  of  Nature,  and  attracts  all 
matter  of  the  other  kind  according  to  the  same  law. 

Two  quantities  of  such  matter  may  be  considered  equal  if, 
when  placed  in  the  same  position  in  a  field  of  force,  they  are 
subjected  to  resultant  forces  which  are  equal  in  intensity  and 
which  have  the  same  line  of  action.  The  two  quantities  of 
matter  are  of  the  same  kind  if  the  direction  of  the  resultant 
forces  is  the  same  in  the  two  cases,  but  of  different  kinds  if  the 
directions  are  opposed.  The  unit  quantity'  of  matter  is  that 
quantity  which  concentrated  at  a  point  would  repel  with  the 
unit  force  an  equal  quantity  of  the  same  kind  concentrated  at 
a  point  at  the  unit  distance  from  the  first  [)oint. 

It  is  evident  from  Articles  2,  14,  and  40  that  m  units  of  one 
of  these  kinds  of  matter,  if  concentrated  at  a  point  (x,  y,  z)  and 
exposed  to  the  action  of  vh,  m.,,  mj,  ...  m„  units  of  the  same 
kind  of  matter  concentrated  respectively  at  the  points  (a'l,  ?/i,2i), 
{x.i,  Pi,  Z.2) ,  (Xs^ys,  z^;),  ...  ix,,y,,  z,),  and  of  ?«*+„  m,^2,  ...  m„ 
units  of  the  other  kind  of  matter  concentrated  respectively  at 
the  points  (a-^+i,  y^^^,  ^t  +  i),  (•i'*-^2i  y*+2^  ^k  +  s),  •••  (^r.^  2/n^2„), 
will  be  urged  in  the  direction  parallel  to  the  positive  axis  of  x 
with  the  force 

X=  -«g?!ki^  +  ,„2?!!il?LpiJl,         [134] 

where   r,-   is   the   distance   between  the   points    {x,  y,  z)    and 
(a:,.,  ?/,.,  2,) . 

If  we  agree  to  distinguish  the  two  kinds  of  matter  from  each 
other  by  calling  one  kind  "  positive  "  and  the  other  kind  ''  neg- 
ative," it  is  easy  to  see  that  if  every  m  which  belongs  to  positive 


78  THE   POTENTIAL   FUNCl^ION 

matter  be  given  the  plus  sign  and  every  m  which  belongs  to 
negative  matter  the  minus  sign,  we  may  write  the  last  equation 
in  the  form 


X 


=  _mV*?^Y^-  [135] 


The  result  obtained  by  making  m  in  [135]  equal  to  unity  is 
called  the  force  at  the  point  (x,  y,  z). 

In  general,  m  units  of  either  kind  of  matter  concentrated  at 
the  point  {x,  y,  z) ,  and  exposed  to  the  action  of  any  continuous 
distribution  of  matter,  will  be  urged  in  the  positive  direction  of 
the  axis  of  x  by  the  force 

V  CCC  p(x'—x)dx'dy'dz'  no^n 

in  this  expression,  p,  the  density  at  {x',y',z'),  is  to  be  taken 
positive  or  negative  according  as  the  matter  at  the  point  is 
positive  or  negative :  m  is  to  have  the  sign  belonging  to  the 
matter  at  the  point  (x,  y,  z)  :  and  the  limits  of  integration  are  to 
be  chosen  so  as  to  include  all  the  matter  which  acts  on  m. 

"With  the  same  understanding  about  the  signs  of  m  and  of  p, 
it  is  clear  that  the  force  which  urges  in  any  direction  s,  m  units 
of  matter  concentrated  at  the  point  (x,y,z)  is  equal  to  —m-D^V, 
where  Fis  the  everywhere  finite,  continuous,  and  single-valued 
function 

p(x'—x)dx'dy'dz' 


///i 


[(x'-xy-^{y'-yy+{z'-zy^i' 

and  that  m  V  measures  the  amount  of  work  required  to  bring  up 
from  "  infinity"  by  any  path  to  its  present  position  the  m  units 
of  matter  now  at  the  point  (cc,  y,  z) . 

If  we  call  the  resultant  force  which  would  act  on  a  unit  of 
positive  matter  concentrated  at  the  point  P  "the  force  at  P," 
it  is  clear  that  if  any  closed  surface  T  be  drawn  in  a  field  of 
force  due  to  any  distribution  of  positive  and  negative  matter  so 
as  to  include  a  quantity  of  this  matter  algebraically  equal  to  Q, 


IN   THE   CASE   OF   REPULSION.  79 

the  surface  integral  taken  over  T  of  the  component  in  the  direc- 
tion of  the  exterior  normal  of  the  force  at  the  different  points  of 
the  surface  is  equal  to  47rQ. 

It  will  be  found,  indeed,  that  all  the  equations  and  theorems 
given  earlier  in  this  chapter  for  the  case  of  one  kind  of  repelling 
matter  may  be  used  unchanged  for  the  case  where  positive  and 
negative  matter  coexist,  if  we  only  give  to  p  and  m  their  proper 
signs. 

It  is  to  be  noticed  that  Poisson's  Equation  is  applicable 
whether  we  are  dealing  with  attracting  matter  or  repelling  mat- 
ter, or  positive  and  negative  matter  existing  together. 

EXAMPLES. 

1.  Show  that  the  extreme  values  of  the  potential  function 
outside  a  closed  surface  S,  due  to  a  quantity  of  matter  algebrai- 
cally equal  to  zero  within  the  surface,  are  its  extreme  values 
on  S. 

2.  Show  that  if  the  potential  function  due  to  a  quantity  of 
matter  algebraically  equal  to  zero  and  shut  in  by  a  closed  sur- 
face S  has  a  constant  value  all  over  the  surface,  then  this  con- 
stant value  must  be  zero. 


80 


SURFACE  DISTRIBUTIONS. 


CHAPTER    IV. 

SUKrAOE  DISTEIBUTIONS.-GEEEN'S  THEOEEM. 

45.  Force  due  to  a  Closed  Shell  of  Repelling  Matter.  If  a 
quantity  of  very  finely-divided  repelling  matter  be  enclosed  in  a 
box  of  any  shape  made  of  indifferent  material,  it  is  evident 
from  [127]  and  from  the  principles  of  Section  38  that  if  the  vol- 
ume of  the  box  is  greater  than  the  space  occupied  by  the  repel- 
ling matter,  the  latter  will  arrange  itself  so  tliat  its  free  surface 
will  be  equipotential  with  regard  to  all  the  active  matter  in 
existence,  taking  into  account  any  there  may  be  outside  tlie  box 
as  well  as  that  inside.  It  is  easy  to  see,  moreover,  that  we 
shall  have  a  shell  of  matter  lining  the  box  and  enclosing  an 
empty  space  in  the  middle. 

That  any  such  distribution  as  that  indicated  in  the  subjoined 
diagram  is  impossible  follows  immediately  from  the  reasoning 
of  Section  37.      For  ABC  and  DEF  are  parts  of  the  same 


Fig.  31. 


equipotential  free  surface  of  the  matter.  If  we  complete  this 
surface  by  the  parts  indicated  by  the  dotted  lines,  we  shall 
enclose  a  space  void  of  matter  and  having  therefore  throughout 
a  value  of  the  potential  function  equal  to  that  on  the  bounding 


green's  theorem.  81 

surface.  But  in  this  case  all  points  which  can  be  reached  from 
0  hy  paths  which  do  not  cut  the  repelling  matter  must  be  at  the 
same  potential  as  0,  and  this  evidently  includes  all  space  not 
actually  occupied  by  the  repelling  matter ;  which  is  absurd. 

Let  us  consider,  then  (see  Fig.  32),  a  closed  shell  of  repelling 
matter  whose  inner  surface  is  equipotential,  so  that  at  every 
point  of  the  cavity  which  the  shell  shuts  in,  the  resultant  force, 
due  to  the  matter  of  which  the  shell  is  composed  and  to  an^- 
outside  matter  there  may  be,  is  zero. 

Let  us  take  a  small  portion  w  of  the  bounding  surface  of  the 
cavity  as  the  base  of  a  tube  of-  force  which  shall  intercept  an 


Fig.  32. 

area  w' on  an  equipotential  surface  which  cuts  it  just  outside  the 
outer  surface  of  the  shell,  and  let  us  apply  Gauss's  Theorem  to 
the  box  enclosed  b\'  ia,  w',  and  the  tube  of  force.  If  F'  is  the 
average  value  of  the  resultant  force  on  w',  the  only  part  of  the 
surface  of  the  box  which  yields  anything  to  the  surface  integral 
of  normal  force,  we  have 

F'oi'  =  4  7r?M, 

where  m  is  the  quantity  of  matter  within  the  box.  If  we  multi- 
ply and  divide  by  w,  this  equation  ma}-  be  written 

F'=^^I^.^.  [137] 

If  o)  be  made  smaller  and  smaller,  so  as  always  to  include  a 
given  point  A^  <o'  as  it  approaches  zero  will  always  inchide  a 
point  B  on  the  line  of  force  drawn  through  A,  and  F'  will  ap- 
proach the  value  F  of  the  resultant  force  at  B. 

The  shell  may  be  regarded  as  a  thick  layer  spread  upon  the 


82  SURFACE  DISTRIBUTIONS. 

inner  surface,  and  in  this  case  the  limit  of  ~  may  be  consid- 

ered  the  vahie  at  A  of  the  rate  at  which  the  matter  is  spread 
upon  the  surface.     If  we  denote  this  limit  by  o-,  we  shall  have 

^=4— iT'of"!  [138] 


If  B  be  taken  just  outside  the  shell,  and  if  the  latter  be  very 

thin,  Jl^'o  [~'i]  evidently  differs  little  from  unity;  and  we  see 

that  the  resultant  force  at  a  point  just  outside  the  outer  sur- 
face of  a  shell  of  matter,  whose  inner  surface  is  equipotential, 
becomes  more  and  more  nearl}^  equal  to  47r  times  the  quantity 
of  matter  per  unit  of  surface  in  the  distribution  at  that  point  as 
the  shell  becomes  thinner  and  thinner. 

The  reader  may  find  out  for  himself,  if  he  pleases,  whether  or 
not  the  line  of  action  of  the  resultant  force  at  a  point  just  out- 
side such  a  shell  as  we  have  been  considering  is  normal  to  the 
shell. 

It  is  to  be  carefully  noticed  that  the  inner  surface  of  a  closed 
shell  need  not  be  equipotential  unless  the  matter  composing  the 
shell  is  finely  divided  and  free  to  arrange  itself  at  will. 

When  the  shell  is  thin,  and  we  regard  it  as  formed  of  matter 
spread  upon  its  inner  surface,  o-  is  called  the  "surface  density" 
of  the  distribution,  and  its  value  at  any  point  of  the  inner  sur- 
face of  the  shell  may  be  regarded  as  a  measure  of  the  amount  of 
matter  which  must  be  spread  upon  a  unit  of  surface  if  it  is  to 
be  uniformly  covered  with  a  layer  of  thickness  equal  to  that  of 
the  shell  at  the  point  in  question. 

46.  Surface  Distributions.  It  often  becomes  necessarj-  in  the 
mathematical  treatment  of  physical  problems,  on  the  assump- 
tion of  the  existence  of  a  kind  of  repelling  matter  or  agent,  to 
imagine  a  finite  quantity  of  this  agent  condensed  on  a  surface 
in  a  layer  so  thin  that  for  practical  purposes  we  may  leave  the 
thickness  out  of  account.  If  a  shell  like  that  considered  in  the 
last  section  could  be  made  thinner  and  thinner  by  compression 


GREEN  S   THEOREM. 


83 


while  the  quantity  of  matter  in  it  remained  iinehanged,  the 
volume  density  (p)  of  the  shell  would  grow  larger  and  larger 
without  limit,  and  o-  would  remain  finite.  A  distribution  like 
this,  which  is  considered  to  have  no  thickness,  is  called  a  sur- 
face distribution. 

The  value  at  a  point  P  of  the  potential  function  due  to 
a  superficial   distribution  of  surface   densit}-  o-  is  the  surface 

integral,  taken  over  the  distribution,  of  -,  where  r  is  the  dis- 

r 
tance  from  P. 

It  is  evident  that  as  long  as  P  does  not  lie  exactly  in  the 
distribution,  the  potential  function  and  its  derivatives  are  always 
finite  and  continuous,  and  the  force  at  any  point  in  any  direc- 
tion may  be  found  by  differentiating  the  potential  function 
partially  with  regard  to  that  direction. 

If  p  were  infinite,  the  reasoning  of  Article  22  would  no 
longer  apply  to  points  actually  in  the  active  matter,  and  it  is 
worth  our  while  to  prove  that  in  the  case  of  a  surface  distri- 
I)ution  where  o-  is  ever^'where  finite,  the  value  at  a  point  i^  of 
the  potential  function  due  to  the  distribution  remains  finite,  as 


P  is  made  to  move  normally  througli  the  surface  at  a  point  of 
finite  curvature. 

To  show  this,  take  the  point  0  (Fig.  33),  where  P  is  to  cut 
the  surface,  as  origin,  and  the  normal  to  the  surface  at  0  as 


84  SUEFACE   DISTRIBUTIONS. 

the  axis  of  x,  so  that  the  other  coordinate  axes  shall  lie  in 
the  tangent  plane. 

If  the  curvature  in  the  neighborhood  of  0  is  finite,  it  will  be 
possible  to  draw  on  the  surface  about  0  a  closed  line  such  that 
for  every  point  of  the  surface  within  this  line  the  normal  will 
make  an  acute  angle  with  the  axis  of  x. 

For  convenience  we  will  draw  the  closed  line  of  such  a  shape 
that  its  projection  on  the  tangent  plane  shall  be  a  circle  whose 
centre  is  at  0  and  whose  radius  is  U,  and  we  will  cut  the  area 
shut  in  b}'  this  line  into  elements  of  such  shape  that  their  pro- 
jections upon  the  tangent  plane  shall  divide  the  circle  just 
mentioned  into  elements  bounded  by  concentric  circumferences 
drawn  at  radial  intervals  of  Am,  and  b}'  radii  drawn  at  angular 
distances  of  A^. 

.If  X,  0,  0  are  the  coordinates  of  the  point  P,  x',  y',  z'  those 
of  a  point  of  one  of  the  elements  of  the  area  shut  in  by  the 
closed  line,  and  a  the  angle  which  the  normal  to  the  surface 
at  this  point  makes  with  the  axis  of  x,  the  size  of  the  surface 

element  is  approximately  - — — —,  where  u^  =  z'^-^  y'^,  and  the 

cosa 

value  at  P  of  the  potential  function  due  to  that  part  of  the  sur- 
face distribution  shut  in  by  the  closed  line  is 

y^  =  rj^  r^ <^d^^     .  [139] 

Jo       Jo     C08a^/{x  —  x'y-i-u^ 

The  quantity 

au  (T  sec  a 


cos  a  -V  {x  —  x'y  +  V? 


M^ 


is  always  finite,  for,  whatever  the  value  of  the  quantity  under 
the  radical  sign  in  the  last  expression  may  be  when  a;,  x',  and  u 
are  all  zero,  it  cannot  be  less  than  unity,  and  therefore  Vi  must 
be  finite  even  when  P  moves  down  the  axis  of  x  to  the  surface 
itself. 

If  V  and  V2  are  the  values  at  P  of  the  potential  functions 
due  respectively  to  all  the  existing  acting  matter  and  to  that 


green's   theorem.  8'J 

part  of  this  matter  not  lying  on  the  portion  of  the  surface  shut 
in  by  our  closed  line,  we  have  F=Fi  +  F2,  and,  since  P  is  a 
point  outside  the  matter  which  gives  rise  to  F,  the  latter  is 
finite  ;  so  that  V  is  finite. 

The  reader  who  wishes  to  studj'  the  properties  of  the  deriva- 
tives of  the  potential  function,  and  their  relations  to  the  force 
components  at  points  actually  in  a  surface  distribution,  will  find 
the  whole  subject  treated  in  the  first  part  of  Riemanu's  ScJiwere, 
Electrkitat  and  Magnetismus. 

Using  the  notation  of  this  section,  it  is  easy  to  write  down 
definite  integrals  which  represent  the  values  of  the  potential 
function  at  two  points  on  the  same  normal,  one  on  one  side  of 
a  superficial  distribution,  and  at  a  distance  a  from  it,  and  the 
other  on  the  other  side  at  a  like  distance,  and  to  show  that  the 
difference  between  these  integrals  may  be  made  as  small  as  we 
like  by  choosing  a  small  enough.  This  shows  that  the  value  of 
the  potential  function  at  a  point  P  changes  continuously,  as  P 
moves  normally  through  a  surface  distribution  of  finite  super- 
ficial density.  If  matter  could  be  concentrated  upon  a  geo- 
metric line,  so  that  there  should  be  a  finite  quantity  of  matter 
on  the  unit  of  length  of  the  line,  or  if  a  finite  quantity  of  matter 
could  be  really  concentrated  at  a  point,  the  resulting  potential 
function  would  be  infinite  on  tlie  line  itself,  and  at  the  point. 

47.  The  Normal  Force  at  Any  Point  of  a  Surface  Distribu- 
tion. In  the  case  of  a  stiietly  superficial  distribution  on  a 
closed  surface  wliere  the  repelling  matter  is  free  to  arrange 
itself  at  will,  the  inner  surface  of  the  matter  (and  hence  the 
outer  surface,  which  is  coincident  with  it)  is  equipotential,  and 
the  resultant  force  at  a  point  B  just  outside  the  distril)ution  is 
normal  to  the  surface  and  numerically  equal  to  4:7  times  the 
surface  density  at  B.  This  shows  that  the  derivative  of  the 
potential  function  in  the  direction  of  the  normal  to  tiie  surface 
has  values  on  opposite  sides  of  the  surface  differing  by  4  -cr. 
and  at  the  surface  itself  cauuot  be  said  to  have  any  definite 
value. 


86  SURFACE  DISTRIBUTIONS. 

It  is  easy,  however,  to  find  the  force  with  which  the  repelling 
matter  composing  a  superficial  distribution  is  urged  outwards. 
For,  take  a  small  element  o)  of  the  surface  as  the  base  of  a  tube 
of  force,  and  apply  Gauss's  Theorem  to  a  box  shut  in  by  the 
surface  of  distribution,  the  tube  of  force,  and  a  portion  w'  of 
an  equipotential  surface  drawn  just  outside  the  distribution. 
Let  F  and  F'  be  the  average  forces  at  the  points  of  w  and  w' 
respectively,  then  the  surface  integral  of  normal  forces  taken 
over  the  box  is  F'w'  —  Fw,  and  this,  since  the  only  active 
matter  is  concentrated  on  the  surface  of  the  box  (see  Section 
31),  is  equal  to  27ro-ow,  where  ct-q  is  the  average  surface  density 
at  the  points  of  the  element  to.     This  gives  us 


F=F'--2Tr(T. 


0* 


Now  let  the  equipotential  surface  of  which  w'  is  a  part  be 
drawn  nearer  and  nearer  the  distribution  ;  then 


lim— =1,    lim  i^' =  47roro,    and    F=2Tra-. 


F  is  the  average  force  which  would  tend  to  move  a  unit  quan- 
tity of  repelling  matter  concentrated  successively  at  the  differ- 
ent points  of  <3i  in  the  direction  of  the  exterior  normal,  but  the 
actual  distribution  on  w  is  wo-p,  so  that  this  matter  presses  on 
the  medium  which  prevents  it  from  escaping  with  the  force 
27ro-o^o);  and,  in  general,  the  pressure  exerted  on  the  resisting 
medium  which  surrounds  a  surface  distribution  of  repelling 
matter  is  at  any  point  27ro^  per  unit  of  surface,  where  cr  is  the 
surface  density  of  the  distribution  at  the  point  in  question. 

We  may  imagine  a  superficial  distribution  of  matter  which  is 
fixed,  instead  of  being  free  to  arrange  itself  at  will.  In  this 
case  the  surface  of  the  matter  will  not  be  in  general  equipoten- 
tial, but,  if  we  apply  Gauss's  Theorem  to  a  box  shut  in  b}-  a 
slender  tube  of  force  traversing  the  disti-ibution,  and  by  two 
surfaces  drawn  parallel  to  the  distribution  and  close  to  it,  one 
on   one   side  and  one  on   the  other,  we  may  prove  that  the 


GIIEEN  S   THEOREM. 


87 


normal  component  of  the  force  at  a  point  just  outside  the  dis- 
tribution differs  by  A-n-a  from  the  normal  component,  in  the  same 
sense,  of  the  force  at  a  point  just  inside  the  distribution  on  the 
line  of  force  which  passes  through  the  first  point. 

48.  Green's  Theorem.  Before  proving  a  very  general  theorem 
due  to  Green,*  of  which  what  we  have  called  Gauss's  Theorem 
is  a  special  case,  we  will  show  that  if  T  is  any  closed  surface 
and  U  a  function  of  .r,  y,  and  z,  which  for  every  point  inside  T 
is  finite,  continuous,  and  single-valued, 

r  r  Cd,U- dxdydz  =  Cu- D^x-ds,  [140] 

where  the  first  integral  is  to  include  all  the  space  shut  in  by  T, 
and  the  second  is  to  be  taken  over  the  whole  surface,  and  where 
Dn  X  represents  the  partial  derivative  of  x  taken  in  the  direction 
of  the  exterior  normal. 

To  prove  this,  choose  the  coordinate  axes  so  that  T  shall  lie 
in  the  first  octant,  and  divide  the  space  inside  the  contour  of  the 


Fir,.  ?A. 


projection  of  T  on  the  plane  yz  into  elements  of  size  dydz.  On 
each  of  these  elements  erect  a  right  prism  cutting  T  twice  or 
some  other  even  number  of  times.  Let  us  call  the  values  of  U 
at  the  successive  points  where  the  edge  nearest  the  axis  of  x  of 

*  George  Green,  An  Essni/  on  the  Application  of  Mnthanntical  Analysis  /<> 
luc  Theories  of  Electricitj  ami  .}fa'jnetism.    Nottingham,  1828. 


80  SURFACE   DISTRIBUTIONS. 

any  one  of  these  prisms  cuts  T,  Ui,  U^i  U3,  ...  f72„  respective!}' ; 
the  angles  which  tiiis  edge  makes  with  exterior  normals  drawn 
to  T  at  these  points,  ai,  aj,  ag,  ...  ao„  V  and  the  elements  which 
the  prism  cuts  from  the  surface  T,  ds^,  ds.j,  ds.,  ...rZsg^.  It  is 
evident  that  wherever  a  line  perpendicular  to  the  plane  yz  cuts 
into  T,  the  corresponding  value  of  a  is  obtuse  and  its  cosine 
negative,  but  wherever  such  a  line  cuts  out  of  T,  the  correspond- 
ing value  of  a  is  acute  and  its  cosine  positive. 

Keeping  this  in  mind,  we  shall  see  that  although  the  base  of 
a  prism  is  the  common  projection  of  all  the  elements  which  it 
cuts  from  T,  and  in  absolute  value  is  approximately  equal  to 
an}'  one  of  these  multiplied  by  the  corresponding  value  of  cos  a, 
yet,  since  dxdy,  dSj,  dsg,  etc.,  are  all  positive  areas  and  some  of 
the  cosines  are  negative,  we  must  write,  if  we  take  account  of  signs, 

dydz  =  — dsi  cos  ai  =  +ds2Cosa2  =  — cZs3Cosa3=  •••. 

If  the  indicated  integration  with  regard  to  x  in  the  left-hand 
member  of  [140]  be  performed  and  the  proper  limits  introduced, 
we  shall  have 

C  C  CD,irdxdydz=C  Cdydz[-  U^-h  U^-  U^+  U, ],  [141] 

where  the  double  sign  of  integration  directs  us  to  form  a  quan- 
tity corresponding  to  that  in  brackets  for  every  prism  which 
cuts  T,  to  multiply  this  b\-  the  area  of  the  base  of  tlie  prism, 
and  to  find  the  limit  of  the  sum  of  all  the  results  ns  the  bases  of 
the  prisms  are  made  smaller  and  smaller. 

Since  we  may  substitute  for  dydz  any  one  of  its  approxi- 
mate values  given  above,  we  may  write  the  quantity  within 
the  brackets 

C7i  cos  tti  dsi  +  Ui  cos  ag  d,%  +  U^  cos  og  ds.^  +  •  •  • , 

and  this  shows  that  the  double  integral  is  equivalent  to  the  sur- 
face integral,  taken  over  the  whole  of  T,  of  ?7cosa,  whence  we 
may  write 

.      C  C  CD,U-dxdydz=  Cucosads,  [142] 


green's  theorem.  89 

where  the  first  integral  is  to  be  taken  all  through  the  space  shut 
iu  by  r,  and  the  second  over  the  whole  surface. 

Let  P{x,  y,  z)  be  an^  point  of  7",  a,  /8,  and  y  the  angles 
which  the  exterior  normal  drawn  to  P  at  T>  makes  with  the 
coordinate  axes,  and  P'  a  point  on  this  normal  at  a  distance 
A/j  from  P.     The  coorclinates  of  P'  are 

x  +  An-cosa,    ?/  + A?( -cos/S,    z -f- An  •  cos  y, 

and  if  W=f{x,y,z)  be  any  continuous  function  of  the  space 
coordinates, 

Wp  =f{x,  y,z), 

Wp'=f(x-{- An  cos  a,  ?/  + An  cos/3,  z-f-Ancosy) 
=f{x,  y,  z)  +  An  cosa  •  D^f+  An  cos^  •  D^f 
and  -f-AncosyZ>,/+(An)2Q, 

IL^-JL£=cosa.DJ-\-cosl3'DJ+cosy-DJ+An.Q, 
whence 
Urn    ^,^/^=£>„  Wp  =  cosaDJ+eosf3DJ-\-cosyDJ.  [143] 

If,  as  a  special  case,  W=x,  we  have  Z>„x=cosa;  so  that 
[1-12]  may  be  written 

C  C  Cd,U-  dxdydz  =  CuD^ix-ds,  [144] 

which  we  were  to  prove.* 

Green's  Theorem,  which  follows  very  easily  from  this  result, 
uiaj'  be  stated  in  the  following  form  : 

If  U  and  V  are  anj'  two  functions  of  the  space  coordinates 
which  together  with  their  first  derivatives  with  respect  to  these 
coordinates  are  finite,  continuous,  and  single-valued  throughout 
the  space  shut  in  by  au}'  closed  surface  7*,  then,  if  a  refers  to 
an  exterior  normal, 

*  This  theorem  has  been  virtually  proved  already  in  Sections  29  and  ;k5. 


90  SURFACE   DISTRIBUTIONS. 

=  Cu-D,V-ds-  C  C  Cu-V-V-dxdydz  [145] 

=  Cv-D„U'ds-  C  C  CV'V'U'dxdydz,  [14G] 

where  the  triple  integrals  include  all  the  space  within  T  and  the 
single  integrals  include  the  whole  surface- 
Since         DM-D,V=D,(U-D,V)-U-D,'V, 

we  have  |    I    i  D^U'D^V-dxdydz 

=  C  C  CDXU-D,V)dxdydz-  C  C  Cu-D^'V-dxdydz; 
but,  from  [144], 

CC  Cd,{ U-D^V)dxdydz  =  Cu-D^V- D^x-ds, 

whence  1    I    I  {DJiJ-D^V)dxdydz 

=  Cu-D,V-D„x-ds-  C  C  Cu-D,^V'dxdydz.    [147] 

If  we  form  the  two  corresponding  equations  for  the  deriva- 
tives with  regard  to  ?/  and  z,  and  add  the  three  together,  we  shall 
obtain  an  expression  which,  by  the  use  of  [143],  reduces  im- 
mediately to  [145].     Considerations  of  symmetr^^  give  [146]. 

If  we  subtract  [146]  from  [145],  we  get 

f  f  f{U-v'V-V'V'U)dxdydz 

=  C{U-D^V-V'D„U)ds.  [148] 

In  applying  Green's  Theorem  to  such  spaces  as  those  marked 
Tq  in  the  adjoining  diagrams,  it  is  to  be  noticed  that  the  walls 
of  the  cavities,  marked  S'  and  S",   as  well  as  the  surfaces, 


green's  theorem.  91 

marked  S,  form  parts  of  the  boundaries  of  the  spaces,  and  that 
the  surface  integrals,  which  the  theorem  declares  must  be  taken 


Fig.  35. 

over  the  whole  boundaries  of  the  spaces,  are  to  be  extended 
over  S'  and  S"  as  well  as  over  S.  We  must  remember,  how- 
ever, that  an  exterior  normal  to  T^  at  S'  points  into  the  cavity  C. 

49.  Special  Cases  under  Green's  Theorem.  I.  If  in  [148] 
V  be  the  potential  function  due  to  any  distribution  either  of 
repelling  matter  or  of  positive  and  negative  matter  existing 
together,  whether  this  matter  is  within  or  without  T,  and  if 
U=l,  we  have 

v'F=-47rp, 

and  47rCCCpdx(1ydz=  C[-D„Vyis.  [149] 

The  triple  integral  on  the  left-hand  side  of  the  equation  is  the 
whole  amount  of  matter  (algebraically  considered,  where  we  have 
both  positive  and  negative  matter)  within  T.  and  the  dexter  is 
the  surface  integral  taken  over  T  of  the  force  in  the  direc- 
tion of  the  exterior  normal;  so  that  [14'J]  expresses  Gauss's 
Theorem. 

II.  If  in  [145]  we  make  6'' equal  to  T^  and  let  this  represent 
as  before  the  potential  function  due  to  any  distriliution  of  actual 
matter  within  or  without  T,  we  shall  have 

f  f  CR-dxdydz  =  Cv- D„  Vds  +  A7rf  f  j'p  Vdxdydz,    [laO] 

where  7i  is  the  resultant  force  at  the  point  (j-, »/,  2). 


92  SUEFACE   DISTRIBUTIONS. 

III.  If  in  [145]  we  make  U=V=u,  any  function  which 
within  the  closed  surface  T  satisfies  the  equation  v^w  =  0,  we 
shall  have 

f  f  r[(^=rW)'+  (D„uy+{D,uy^  dxdydz  =  Cu -D^u-  ds.  [151] 

IV.  If  in  [148]  Fis  the  potential  function  due  to  two  distri- 
butions of  active  matter,  Mi  inside  the  closed  surface  T  and  M^ 

outside  it,  and  if  U=  -  where  r  is  the  distance  of  the  point 

(x,  y,  z)  from  a  fixed  point  0,  we  must  consider  separately  the 
two  cases  where  0  is  respectively  without  T  and  within  T. 

A.  If  0  is  without  T,  V^  (  -  j  =  0  for  points  within  the  sur- 
face.    Also,  V  F=  — 47rp,  so  that 

f^ds  -Jv-  D,  (i)  ds  =  -iTrfff  ^-dxdydz. 


Fig.  36. 


The  triple  integral  is  evidently  equal  to  the  value  at  the  point 
0  of  the  potential  function  due  to  Mi  alone.  If  we  call  this  Fi, 
and  notice  (see  [143])  that  I)„r  at  any  point  of  T  is  the  cosine 
of  the  angle  8  between  r  and  the  exterior  normal  to  T,  we  have 

JD^cls-f^ds=-i.n  [152] 

If  r  is  a  surface  eqnipotential  with  respect  to  the  joint  action 
of  Ml  and  M.,,  and  if  we  denote  by  F,  the  constant  value  of  V 
on  r,  we  have 

I  — =—  ds  —  V,l  — — -  ds  =  —  4 TT  Fi, 


green's  theorem.  93 

and  it  is  easy  to  show,  by  the  reasoning  used  in  Section  31, 

that   I    '        ds  =  0,  whence 
J     r- 

Fi  =  -  —  f^^ds.  [153] 

4:'7rJ      r 

B.  If  0  is  a  point  inside  T,  whether  or  not  it  is  within  3/,, 
and  if  T  is  equipotential  with  respect  to  the  action  of  M^  and 
3/;,  we  will  surround  0  b}'  a  small  spherical  surface  s'  of 
radius  r',  and  apply  [148]  to  the  space  lying  inside  T  and  with- 
out the  spherical  surface.  In  doing  so,  it  is  to  be  noticed  that 
s'  forms  part  of  the  boundary  of  the  region  we  are  dealing  with, 
and  that  an  exterior  normal  to  the  region  at  s'  will  be  an  interior 
normal  of  the  sphere. 


Fig.  37. 


we  have 


Since  for  all  points  of  the  region  we  arc  considering  V*[  -  |  =  0. 
\  have 

=  -4 Trjyj"^  dxdydz  ;  [154] 

or,  since  ds'=r'-d<i}\  where  dui'  is  the  area  which  the  elementary 
cone  whose  base  is  ds'  and  vertex  0  intercepts  on  the  sphere 
of  unit  radius  drawn  about  0, 


94  SURFACE   DISTRIBUTIONS. 

.    It  is  easily  proved,  by  the  reasoning  of  Section  31,  that 
rcosS 


J  cos 6    ,  , 


and  it  is  clear  that  if  r'  be  made  smaller  and  smaller,  the  third 
integral  of  [155]  approaches  the  limit  zero.  If  V'  is  the  average 
value  of  Fon  the  surface  s', 


jV'da)=F'rda;  =  "F'47r; 


and  as  r'  is  made  smaller  and  smaller,  this  approaches  the  value 
AttVoi  where  Vq  is  the  value  of  Fat  0.  The  value,  when  r'  is 
zero,  of  the  triple  integral  in  [155]  is  evidently  Fi,  and  we 
have 

'^!Lrds  +  47rF.-47rFo  =  -47rFi.  [156] 

r 


/ 


If  F2  is  the  value  at  0  of  the  potential  function  due  to  M^ 
alone,  Fq  =  Fi  +  F2,  so  that  [156]  may  be  written  in  the  form 

F,  -  F2  =  -  /-  C^^ds.  [157] 

If  T  is  not  equipotential  with  respect  to  the  action  of  M^  and 
Jfj,  we  have 

4  TT  V2  ^C^ds  -f  VD„  f-\  ds.  [158] 

V.    If  in  [148]  we  make  U=-,  where  r  is  the  distance  of 

the  point  (x,y,z)  from  a  fixed  point  0,  and  if  V=v,  a  function 
which  within  the  closed  surface  T  satisfies  the  equation  V'^'y  =  0, 
we  shall  have 

4  TTV  =  CvD„  f-^  ds  -  r^  ds,  [159] 

if  0  is  within  T,  and 

C^  ds  =  Cv  ■  D„  (^  ds,  [160] 

if  0  is  outside  T. 


green's  theorem.  95 

50.  Th3  Surface  Distributions  Equivalent  to  Certain  Volume 
Distributions.  Keeping  the  notation  of  IV.  in  tlie  last  article, 
let  r  be  a  closed  surface  equipotential  with  respect  either  to 
the  joint  action  of  two  distributions  of  matter,  J/i  inside  T  and 
J/2  outside  it,  or  (when  3/,  equals  zero)  to  the  action  of  a 
single  distribution  within  the  surface  ;  and  let  Fj,  V^,  and  V 
be  the  values  of  the  potential  functions  due  respectively  to  M^ 
alone,  to  My  alone,  and  to  My  and  M^  existing  together.  If  a 
quantity  of  matter  were  condensed  on  T  so  as  to  give  at  everv 

—  D  V 

point  a  surface  deusitv  equal  to '^—,  the  whole  quantity  of 

47r 
matter  on  the  surface  would  be 


-  CD„V'ds, 

IT  J 


4: 

and  this,  by  [H9],  is  equal  in  amount  to  3/j.    Let  us  study  the 

effect  of  removing  M^  from  the  inside  of  T  and  spreading  it  in 

a  superficial  distribution  J/,'  over  T,  so  that  the  surface  density 

—  1)  V 
ut  every  point  shall  be '— —     In  what  follows,  it  is  assumed 

that  we  have  two  distributions  of  matter,  one  inside  the  closed 
surface  and  the  otiier  outside.  It  is  to  be  carefulh"  noted,  how- 
ever, that  by  putting  3/o  equal  to  zero  in  our  equations,  all  our 
results  are  applicable  to  the  case  where  we  have  an  equipotential 
surface  surrounding  all  the  matter,  which  may  be  all  of  one  kind 
or  not. 

The  value,  at  any  point  0,  of  the  potential  function  due  to 
the  joint  effect  of  Mo  and  the  surface  distribution  J//,  would  l)e 

4  IT  J      r 
If  0  is  an  outside  point,  we  have,  by  [153], 
Vo  =  V,  +  V,. 

so  that  the  effect  at  any  point  outside  an  equipotential  surface 
of  a  quantity  My  of  matter  anyhow  distrilmted  inside  the  sur- 
face is  the  same  as  tliat  of  an  equal  quantity  of  matter  dis- 
tributed over  the  surface   in   such  a  way  that  the   superficial 


96  SURFACE   DISTRIBUTIONS. 

density  at  ever}-  point  is  ^^ — "- ,  where  V  is  the  value  of  the 

potential  function  clue  to  the  joint  action  of  Jfj  and  any  matter 
{M2)  that  may  be  outside  the  surface. 
If  0  is  an  inside  point,  we  have,  by  [157], 

Vo=V,-hV,-V2  =  V„  [161] 

which  shows  that  the  joint  effect  of  M2  and  3/i'  is  to  give  to  all 
points  within  and  upon  the  surface  the  same  constant  value  of 
the  potential  function  which  points  upon  the  surface  had  before 
Ml  was  displaced  by  M^'.  If,  therefore,  Jf/  and  3/2  exist  without 
Ml,  there  is  no  force  at  any  point  of  the  cavity  shut  in  by  T; 
or,  in  other  words,  the  force  due  to  Mi  alone  is  at  all  points 
inside  T  equal  and  opposite  to  that  due  to  3/2- 

If  Ml  and  M^  exist  without  Mi,  the  cavity  enclosed  by  T  is,  in 
general,  a  field  of  force.  Mi  acts  as  a  screen  to  shield  the  space 
within  T  from  the  action  of  Mo. 

The  surface  of  Mi  is  equipotential  with  respect  to  all  the 
active  matter,  so  that  there  is  no  tendency  of  the  matter  com- 
posing the  surface  distribution  to  arrange  itsfelf  in  any  different 
manner  upon  T. 

51.  The  potential  function  F,  due  to  any  distribution  of 
matter  whose  volume  density  p  is  everywhere  finite,  satisfies  the 
following  conditions : 

(1 )  V  and  its  first  space  derivatives  are  everywhere  finite  and 
continuous,  and  are  equal  to  zero  at  an  infinite  distance  from 
the  attracting  mass. 

(2)  If  a  is  the  distance  from  the  origin  of  coordinates  to  the 
point  P, 

where  M  is  a  definite  constant. 

(3)  Except  at  the  surface  of  the  attracting  mass,  or  at  some 
other  surface  where  p  is  discontinuous,  . 

vV=-47rp, 
where  p  is  to  be  put  equal  to  zero  outside  of  the  attracting  mass. 


green's  theoeem.  97 

It  is  easy  to  show  from  Green's  Theorem  that  for  a  given 
value  of  p  as  a  function  of  a*,  ?/,  and  z,  only  one  function  which 
will  satisfy  these  three  conditions  exists. 

Suppose,  for  the  sake  of  argument,  that  there  are  two  such 
functions,  Fand  F',  and  put  u  =  F—  F'.  It  is  evident  that  u 
satisfies  conditions  (1)  and  (2),  and  that  V"(m)  =  0  except  where 
p  is  discontinuous.  Parallel  to  each  surface  of  discontinuity, 
and  very  near  to  it,  draw  two  surfaces,  one  on  each  side,  so  as 
to  shut  in  the  places  where  V«  is  not  zero,  and  draw  a  spherical 
surface  about  the  origin,  using  a  radius  R  large  enough  to 
eiK'lose  all  the  surfaces  of  discontinuity. 

If  now  we  apply  [lol]  to  that  part  of  the  space  inside  the 
spherical  surface  and  not  shut  in  by  the  barriers  which  we  have 
drawn,  and  if  we  notice  that  each  pair  of  parallel  barriers  to- 
gether yields  nothing  to  the  surface  integral,  we  shall  have 

^ ^ j\_{D,uy  +  {D,uy  +  {D,uy-\dxdydz^  ju.  D^u.ds, 

where  the  dexter  integral  is  to  be  extended  over  the  spherical 
surface  only. 

If  dia  is  the  solid  angle  of  the  infinitesimal  cone  which  inter- 
cepts the  element  ds  from  the  spherical  surface,  we  have 


I  u  X>„  u  ds  =  H-  I  u  Dji  u  do 


Now  since  ?«  satisfies  condition  (2)  above,  it  is  easy  to  show 
that  if  we  make  li  grow  larger  and  larger,  this  surface  integral 
approaches  the  value  zero  as  a  limit,  for  u  approaches  the  value 

^^  and  Dj,u  the  value   ~  ^    .    so  that  the   wliole   integral  ap- 

proaches  the  value  ^^^^ — — — ,  which,  when  A'  i.s  made  infinite, 

approaches  the  value  zero.  , 

If  we  em])race  all  space  in  our  sphere,  we  shall  have 


///< 


(Z),?0-  +  (/>,")'  +  (^x«)']  dxdiidz  =  0, 
whence  D^  n  =  0,     Z>,  u  =  0,     Z>,  m  =  0. 


98  SURFACE  DISTRIBUTIONS. 

Therefore  u  is  constant  in  all  space,  and  since  it  is  zero  at 
infinity,  mnst  be  everywhere  zero,  so  that  V=  V. 

^3^  52.  Thomson's  Theorem  or  Dirichlet's  Principle.  We  will  now 
prove  a  theorem*  which  is  usually  called  Dirichlet's  Principle 
by  Continental  writers,  but  which  in  English  books  is  generally 
attributed  to  Sir  W.  Thomson.  This  theorem,  in  its  simplest 
form,  asserts  that  there  always  exists  one,  but  no  other  than 
this  one,  function,  v,  of  x,  y,  z,  which  (1)  is  finite,  continuous,  and 
single-valued,  together  with  its  first  space  derivatives,  through- 
out a  given  closed  region  L ;  (2)  at  every  point  of  the  region 
satisfies  the  equation  v'v  =  0;  and  (3)  at  every  point  on  the 
boundary'  of  the  region  has  any  arbitrarily  assigned  value,  pro- 
vided that  this  can  be  regarded  as  the  value  at  that  point  of  a 
single-valued  function  which  has  derivatives  finite,  continuous, 
and  single-valued  all  over  this  boundary. 

Tliere  is  evidently  an  infinite  number  of  functions  which 
satisfy  the  first  and  third  conditions.  If,  for  instance,  the  equa- 
tion of  the  bounding  surface  S  of  the  region  is  -F'(.^^  y,  z)  =  0, 
and  if  the  value  of  v  at  the  point  (a;,  y,  z)  upon  this  surface  is  to 
hef{x,y,z),  any  function  of  the  form 

<^(x,  y,  z)  'F{x,  y,  z)  -\-f{x,  y,  z) 

would  satisfy  the  third  condition,  whatever  finite  function  $ 
might  be. 

If  we  assign  to  the  function  to  be  found  a  constant  value  C 
all  over  S,  v^C  will  satisfy  all  three  of  the  conditions  given 
above. 

*  Green,  An  Essay  on  the  Application  of  Mathematical  Analysis  to  the 
Theories  of  Electricity  and  Magnetism.  Gauss,  AUgemeine  Lelirsutze  in  Dezie- 
huntj  auf  die  im  verkehrten  Verhdltnisse  des  Quadrats  der  Entfernung  wirkcnden 
Anziehunys-  und  Abstossungsh-iifle.  Thomson,  Reprint  of  Papers  on  Electro- 
statics and  Magnetism.  Dirichlct,  Vorlesungen  iiher  die  im  umgekehrten  Ver- 
hdltniss  des  Quadrats  der  Entfernung  ivirkenden  Krafte.  Also,  Thomson  and 
Tail's  Natural  Piiiitisnpliy,  and  several  papers  by  Dirichlet  in  Crelle's  Jour- 
nal and  in  the  Comptes  Rendus. 


green's  theorem.  99 

If  the  sought  function  is  to  have  different  values  at  different 
points  of  S,  and  if  for  u  in  the  integral 

Q  =/jy[(^^«)'  +  (^^v«)'  +  (^x^)']  dxdydz, 

which  is  to  be  extended  over  the  whole  of  the  region,  we  substi- 
tute any  one  of  all  the  functions  wliich  satisfy  conditions  (1) 
and  (3),  the  resulting  value  of  Q  will  be  positive.  Some  one  at 
least  of  these  functions  (y)  must,  however,  yield  a  value  of  Q 
which  though  positive,  is  so  small  that  no  other  one  can  make  Q 
smaller.  Let  h  be  an  arbitrary  constant  to  which  some  value 
has  been  assigned,  and  let  vj  be  any  function  which  satisfies 
condition  (1)  and  is  equal  to  zero  at  all  parts  of  S,  then 
U=v-\-hic  will  satisfy  conditions  (1)  and  (3),  and  conversely, 
there  is  no  function  which  satisfies  these  two  conditions  which 
cannot  be  written  in  the  form  U=v  -\-hw^  where  h  is  an  arbi- 
trary constant,  and  tv  a  function  which  is  zero  at  S  and  which 
satisfies  condition  (1). 

Call  the  minimum  value  of  Q  due  to  v^  Q,,  and  the  value  of  Q 
due  to  U,  Qui  then 

Qu  =  Q„+'2h  C  C  C{D^v-D^w  +D^v-D^w  +  D,v •  D,v:)dxdydz 

+  h'  fff[{D^icy--h  {D.ioy  +  [D^icy-]  dxdydz, 

which,  since  to  is  zero  at  the  boundary  of  the  region,  may  be 
written,  by  the  help  of  Green's  Theorem, 

Q^.  -  Q„  =  -  2  /«  r  f  Cw  v' ( I')  dx dy  dz  -\-  h-  n\ 

Now  since  Q^  is  the  minimum  value  of  Q,  no  one  of  the  infi- 
nite number  of  values  of  Qc-  —  Qv  formed  by  changing  h  and  iv 
under  the  conditions  just  named  can  be  negative  ;  but  if  V'u 
were  not  everywhere  equal  to  zero  within  L,  it  would  be  easy 
to  choose  ?o  so  that  the  coefficient  of  '2h  in  the  expression 
for  Qu—  Q„  should  not  be  zero,  and  then  to  choose  /*  so  that 
Qu~Qy>  should  be  negative.    Hence  V^v  is  equal  to  zero  through- 


100  SURFACE  DISTRIBUTIONS. 

out  i,  and  there  alwa3's  exists  at  least  one  function  whicli  satis- 
fies the  three  conditions  stated  above. 

Tliere  is  onl}'  one  such  function  ;  for  if  beside  v  there  were 
another  m  =  v  -f-  ^^'^i  we  should  have,  since  the  coefficient  of  h  is 
zero  when  V^(w)  =  0, 

and,  that  Q„  may  be  as  small  as  Q„,  liQ,  must  be  zero,  whence 
either  /i  =  0  or  O  =  0,  and  if  O  =  0,  w  is  zero.  Therefore, 
u  =  v,  and  there  is  only  one  function  which  in  any  given  case 
satisfies  all  the  three  conditions  given  above. 

By  applying  the  same  reasoning  to  the  space  outside  a  closed 
surface  S  and  inside  a  spherical  surface  of  large  radius  R  which 
is  finally  made  infinite,  it  is  easy  to  j^rove  that  there  always  exists 
in  the  space  outside  a  closed  surf  ace  aS' one  and  only  one  function 
V  which  (1)  has  a  given  value  at  every  point  of  S,  (2)  satisfies 
the  equation  v'^v  =  0,  (3)  together  with  its  first  derivatives,  is 
finite  and  continuous  outside  S.,  and  (4)  is  such  that  the  limit, 
as  R  becomes  infinite,  of  Rv  is  a  definite,  finite  constant. 

These  theorems  help  us  to  prove  other  theorems,  of  which  two 
are  of  considerable  interest  for  us. 

I.  If  a  function  v=f{x,y,z),  together  with  its  first  space 
derivatives,  is  finite  and  continuous  in  all  space  outside  a  sur- 
face S,  and  outside  this  surface  satisfies  the  equation  V'f  =  0, 
and  if  v-\/x^  +  y^  +  z^  approaches  a  definite,  finite,  constant 
limit  as  .the  point  (cc,  y,  z)  moves  away  from  the  origin  to  in- 
finity, then  this  function  may  be  considered  to  be  the  potential 
function  of  a  surface  distribution  of  matter  upon  S. 

In  order  to  prove  this,  we  will  first  apply  [160]  to  v',  the 
function  which  has  on  S  the  same  value  as  v,  which  inside  8  is, 
with  its  first  derivatives,  finite  and  continuous,  and  which  satisfies 
the  equation  V'v'=  0  ;  and  use  the  space  inside  S  as  our  region. 
This  gives 

where  n  refers  to  the  exterior  normal  of  S. 


green's  theorem.  101 

If  we  now  apply  [150]  to  the  function  v,  using  as  a  field  tlic 
space  outside  S  and  within  a  spherical  surface  *S"  of  large  radius 
i?,  drawn  about  the  point  0  as  centre,  we  shall  have 

where  n  is  made  to  refer  to  the  same  normal  as  before  by  a 

change  of  sign  in  the  first  two  integrals.     If  now  we  combine 

the  two  equations  just  obtained,  and  make  R  infinite,  so  that  the 

last  two  integrals  of  the  second  equation  shall  vanish,  we  shall 

have 

ds 

r 


which  is  the  value  at  an  outside  point  of  the  potential  function  due 
to  a  superficial  distribution  of  surface  density  — (D„v'~D„v) 
spread  upon  S. 

It  is  to  be  noticed  that  the  letter  r  refers  to  a  point  without 
S  in  each  of  the  last  three  equations.  Instead  of  one  closed 
surface  we  might  have  several,  as  it  is  eas}-  to  prove  by  intro- 
ducing as  many  Dirichlet's  functions  as  there  are  surfaces. 

We  will  state  the  second  theorem,  leaving  the  proof,  wl^ch  is 
almost  identical  in  form  with  the  one  just  given,  for  the  reader. 

II.  If  a  function  v'=F(x,y,z)  satisfies  the  equation  V'i''=  0 
throughout  the  space  enclosed  by  a  closed  surface  S,  and  within 
this  space,  together  with  its  first  derivatives,  is  everywhere 
finite  and  continuous,  it  may  be  considered  to  be  the  potential 
function  within  this  space  of  a  surface  distribution  on  .S'. 

The  superficial  density  of  tliis  distribution  will  be  found  to  1)0 

4  — 
where  v  is  the  function  which  has  the  same  value  on  S  that  *•' 
has,  and  outside  S  satisfies  the  equation  vX'")  =  0  ^^"^1  the  other 
conditions  given  above. 

It  follows,  from  these  theorems,  that  we  may  assign  any  con- 
tiuuouslv  arranged  arbitrarv  values  to  the  potential  function  at 


102  STJEFACE   DISTRIBUTIONS. 

the  different  points  of  a  closed  surface  S,  make  these  values  the 
common  values  on  the  surface  of  the  functions  v  and  v',  and 
assert  that  a  distribution  of  matter  on  >S'  of  surface  density 

o- =  — \DnV' —  DnV)  would  give   rise  to  a  potential  function 

having  the  chosen  values  on  S.  In  this  case  v  and  v'  would  be 
the  values  in  regions  respectively  without  and  within  S  of  the 
potential  function  due  to  this  surface  distribution.  It  is,  then, 
always  possible  to  distribute  matter  in  one  and  onl}'  one  way 
upon  a  given  closed  surface  so  that  the  value  of  the  potential 
function  due  to  the  matter  shall  have  given  values  all  over  the 
surface. 

EXAMPLES. 

1.  Prove  that  there  always  exists  one,  but  no  other  than  this 
one,  function,  v,  which,  together  with  its  first  space  derivatives, 
is  finite,  continuous,  and  single-valued  everywhere  within  a  given 
region  L,  has  values  at  the  boundary  of  the  region  equal  to 
those  of  an  arbitrarily  chosen,  finite,  continuous,  and  single- 
valued  function, /(ic,  y,  2),  and  satisfies  at  every  point  in  L  the 
equation 

D,{K-  D^v)  +  D^{K-  D^v)  +  D,(K-  D,v^  =  0, 

where  K  is  a  function  positive  within  L. 

2.  If  the  potential  function  due  to  a  certain  distribution  of 
matter  is  given  equal  to  zero  for  all  space  external  to  a  given 
closed  surface  8  and  equal  to  «^(a;,  y^  2;),  where  <^  is  a  continu- 
ous single-valued  function  zero  at  all  points  of  >S',  for  all  space 
within  S  ;  there  is  no  matter  without  S.,  there  is  a  superficial  dis- 
tribution of  surface  density 

cr  =  J-[(A<A)^  +  (^.^)='  +  (^.<^)T 

47r 

upon  aS,  and  the  volume  density  of  the  matter  within  S  is 

[Thomson  and  Tait.] 


ELECTROSTATICS.  103 


CHAPTER   V. 
ELEOTEOSTATIOS. 

53.  Introductory.  Having  considered  abstractly  a  few  of 
the  characteristic  properties  of  what  has  been  called  "the  New- 
tonian potential  function,"  we  will  devote  this  chapter  to  a  very 
brief  discussion  of  some  general  principles  of  Electrostatics. 
By  so  doing  we  shall  incidentally  learn  how  to  apply  to  the  treat- 
ment of  practical  problems  many  of  the  theorems  that  we  have 
proved  in  the  i)receding  chapters. 

In  what  follows,  the  reader  is  supposed  to  be  familiar  with 
such  electrostatic  phenomena  as  are  described  in  the  first  few 
chapters  of  treatises  on  Statical  Electricity,  and  with  the  hypoth- 
eses that  arc  given  to  explain  these  phenomena. 

Without  expressing  any  opinion  with  regard  to  the  physical 
nature  of  what  is  called  electrification^  we  shall  here  take  for 
granted  that  whether  it  is  due  to  the  presence  of  some  sub- 
stance, or  is  only  the  consequence  of  a  mode  of  motion  or  of  a 
state  of  polarization,  we  may,  without  error  in  our  results,  use 
some  of  the  language  of  the  old  ''Two  Eluid  Theory  of  Elec- 
tricity "  as  the  basis  of  our  mathematical  work. 

The  reader  is  reminded  that,  among  other  things,  this  theory 
teaches  that :  — 

(1)  Every  particle  of  a  body  which  is  in  its  natural  state  con- 
tains, combined  together  so  as  to  cancel  each  other's  etfects  at 
all  outside  points,  equal  large  quantities  of  two  kinds  of  elec- 
triciti/  with  properties  like  those  of  the  positive  and  negative 
"  matter"  described  in  Section  44. 

(2)  Electrification  consists  in  destroying  in  some  way  the 
equality  between  the  amounts  of  the  two  kinds  of  electricity 
which  a  body,  or  some  part  of  a  body,  naturally  contains,  so 
that  there   shall  be  an   excess  or  chanje  of  one  kind.     If  the 


104  ELECTROSTATICS. 

charge  is  of  positive  electricity,  the  body  is  said  to  be  posi- 
tivel}'  electrified  ;  if  tlie  charge  is  negative,  negatively  electrified. 
Either  kind  of  electricity  existing  uncoinbined  Avith  an  equal 
quantity  of  the  other  kind,  is  called  free  electricity. 

(3)  When  a  charged  body  A  is  brought  into  the  neighborhood 
of  another  body  B  in  its  natural  state,  the  two  kinds  of  elec- 
tricity in  every  particle  of  B  tend  to  separate  from  each  other, 
one  being  attracted  and  the  other  repelled  by  ^'s  charge,  and 
to  move  in  opposite  directions. 

In  general,  a  tendency  to  separation  occurs  in  all  parts  of  the 
body,  whether  it  is  charged  or  not,  where  the  resultant  electric 
force  (the  force  due  to  all  the  free  electricity  in  existence)  is 
not  zero.     This  effect  is  said  to  be  due  to  induction. 

In  our  work  we  shall  assume  all  this  to  be  true,  and  proceed 
to  apply  the  principles  stated  in  Section  44  to  the  treatment  of 
problems  involving  distributions  of  electricit}-.  We  shall  find  it 
convenient  to  distinguish  between  conductors.,  which  offer  prac- 
tically no  resistance  to  the  passage  of  electricity  through  their 
substance,  and  nonconductors,  which  we  shall  regard  as  prevent- 
ing altogether  such  transfer  of  electricity  from  part  to  part. 

54.  The  Charges  on  Conductors  are  Superficial.  When  elec- 
tricity is  communicated  to  a  conductor,  a  state  of  equilibrium  is 
soon  established.  After  this  has  taken  place,  there  can  be  no 
resultant  force  tending  to  move  any  portion  of  the  charge 
through  the  substance  of  the  conductor,  for,  by  supposition,  the 
conductor  does  not  prevent  the  passage  of  electricity  through 
itself. 

Moreover,  the  resultant  electric  force  must  be  zero  at  all 
points  in  the  substance  of  a  conductor  in  electric  equilibrium ; 
for  if  the  force  were  not  zero  at  any  point,  electricity  would 
be  produced  by  induction  at  that  point,  and  carried  away 
through  tlie  body  of  the  conductor  under  the  action  of  the 
inducing  force. 

From  this  it  follows  that  the  potential  function  V.  due  to  all 
the  free  electricity  in  existence,  must  be  constant  throughout 


ELECTROSTATICS.  105 

the  substance  of  an}*  single  conductor  in  electric  equilibrium, 
wlietber  or  not  the  conductor  be  charged,  and  whether  or  not 
there  be  other  charged  or  uncharged  conductors  in  the  neigh- 
borhood. Different  conductors  existing  together  will  in  general 
be  at  diflferent  potentials,  but  all  the  points  of  any  one  of  these 
conductors  will  be  at  the  same  potential. 

"Wherever  V  is  constant,  V'I'=0,  and  hence,  by  Poisson's 
Equation,  p  =  0,  so  that  there  can  be  no  free  electricity  within 
the  substauce  of  a  conductor  in  equilil)rium,  and  the  whole 
charge  must  be  distributed  upon  the  surface.  Experiment 
shows  that  we  must  regard  the  thickness  of  charges  spread  upon 
conductors  as  inappreciable,  and  that  it  is  best  to  consider  tliat 
in  such  cases  we  have  to  do  with  really  superficial  distributions 
of  electricity,  in  which  the  conductor  bears  a  rough  aualogv  to 
the  cavity  enclosed  by  the  thin  shells  of  repelling  matter  de- 
scribed in  the  preceding  chapter. 

The  surface  density  at  any  point  of  a  superficial  distribution 
of  electricity  shall  be  taken  positive  or  negative,  according  as 
the  electricity  at  that  point  is  positive  or  negative,  and  the  force 
which  would  act  upon  a  unit  of  positive  electricity  if  it  were 
concentrated  at  a  point  J*  without  disturbing  existing  distribu- 
tions shall  be  called  '"tlie  electric  force"  or  "the  strength  of 
the  electric  field  at  P." 

It  is  evident,  from  Sections  4.j  and  40,  that  the  electric  force 
at  a  point  just  outside  a  charged  conductor,  at  a  place  where 
the  surface  density  of  the  charge  is  o-,  is  47r<r,  and  that  this  is 
directed  outwards  or  inwards,  according  as  a-  is  positive  or  nega- 
tive. 

In  other  words,  D^V,  the  derivative  of  the  potential  function 
in  the  direction  of  the  exterior  normal,  is  equal  to  —  4  Tnr,  and 
the  value  of  Fat  a  point  P  just  outside  tlie  conductor  is  greater 
or  less  than  its  value  within  the  conductor,  according  as  the 
surface  density  of  the  conductor's  charge  in  tlie  neighborhood  of 
P  is  negative  or  positive. 

It  is  to  be  carefully  noted  that,  although  the  surface  of  a  con- 
ductor must  always  be  equipotential,  the  superficial  density  of 


106  ELECTilOSTx\.TlC3. 

the  conductor's  charge  need  not  be  the  same  at  all  parts  of  the 
surface.  We  shall  soon  meet  with  cases  where  the  electricity 
on  a  conductor's  surface  is  at  some  points  positive  and  at  others 
negative,  and  with  other  cases  where  the  sign  of  the  potential 
function  inside  and  on  a  conductor  is  of  opposite  sign  to  the 
charge. 

It  is  evident,  from  the  work  of  Section  47,  that  the  resistance 
per  unit  of  area  which  the  nonconducting  medium  about  a  con- 
ductor has  to  exert  upon  the  conductor's  charge  to  prevent  it 
from  flying  off,  is,  at  a  part  where  the  density  is  cr,  27rcr-. 

55.  General  Principles  which  follow  directly  from  the  Theory 
of  the  Newtonian  Potential  Function.  If  two  different  distrilju- 
tions  of  electricity,  which  have  the  same  system  of  equipoten- 
tial  surfaces  throughout  a  certain  region,  be  superposed  so  as  to 
exist  together,  the  new  distribution  will  have  the  same  equipo- 
tential  surfaces  in  that  region  as  each  of  the  components.  For, 
if  Vi  and  V»,  the  potential  functions  due  to  the  two  components 
respectively,  be  both  constant  over  any  surface,  their  sum  will 
be  constant  over  the  same  surface. 

Two  distributions  of  electricity,  which  have  densities  ever}'- 
where  equal  in  magnitude  but  opposite  in  sign,  have  the  same 
system  of  equipotential  surfaces,  and,  if  superposed,  have  no 
effect  at  any  point  in  space. 

Two  distributions  of  electricity-,  arranged  successively  on  the 
same  conductor  so  that  at  every  point  the  density  of  the  one 
is  VI  times  that  of  the  other,  have  the  same  system  of  equipo- 
tential surfaces,  and  the  potential  function  due  to  the  first  is 
everywhere  m  times  as  groat  as  that  due  to  the  second. 

If  the  whole  charge  of  a  conductor  which  is  not  exposed  to 
the  action  of  any  electricity  except  its  own  is  zero,  the  super- 
ficial density  must  be  zero  at  all  points  of  the  surface,  and  the 
conductor  is  in  its  natural  state.  For  if  o-  is  not  everywhere 
zero,  it  must  be  in  some  places  positive  and  in  others  negative  ; 
and,  according  to  the  work  of  the  last  section,  the  potential 
function  F,  due  to  this  charge,  must  have,  somewhere  outside 


ELECTROSTATICS.  107 

the  conductor,  values  higher  and  lower  than  T\,,  its  value  in  the 
conductor  itself.  But  this  would  necessitate  somewhere  in  empty 
space  a  value  of  the  potential  function  not  lying  between  Vq  and 
0,  the  value  at  infinity  ;  that  is,  a  niaxinuun  in  empty  space  if 
T'o  is  positive,  and  a  minimum  if  T'o  is  negative ;  which  is 
absurd. 

A  system  of  conductors,  on  each  of  which  the  charge  is  null, 
must  be  in  the  natural  state  if  exposed  to  the  action  of  no  out- 
side electricity.  For,  by  applying  the  reasoning  just  used  to 
that  conductor  in  which  the  potential  function  is  supposed  to 
have  the  value  most  widely  different  from  zero,  we  m.ay  show 
that  tlie  surface  density  all  over  the  conductor  is  zero,  so  that 
no  influence  is  exercised  on  outside  bodies  ;  and  then,  snp[)os- 
ing  this  conductor  removed,  we  may  proceed  in  the  same  way 
with  the  system  made  up  of  the  remaining  conductors. 

If  a  charge  M  of  electricity,  wljen  given  to  a  conductor,  ar- 
ranges itself  in  equilibrium  so  as  to  give  the  surface  density 

o-  =/(.i', !/'  z)  and  to  make  the    potential   function  Fq  =  j  '^— ^ 

« '     r 

constant  within  the  conductor,  ^  cliarge  —JF,  if  arranged  on  the 
conductor  so  as  to  give  at  every  point  the  density  —<t=  —f{x,i/,z) 
would  be  in  equilibrium,  for  it  would  give  everywhere  the  poten- 
tial function   I  ~J^1^ — L  =  —  1"^,  and  this  is  constant  wherever  I'^ 

J        r 
is  constant. 

Only  one  distribution  of  the  same  quantity  of  electricity  ^[  on 
the  same  conductor,  removed  from  the  influence  of  all  other 
electricity,  is  possible  ;  for,  suppose  two  different  values  of  sur- 
face density  possible,  (Ti=/i{x,  y,z)  and  'r^  =^>  (.r, »/, :;),  then 
—  <r2= —f:>{x.y,z)  is  a  possible  distribution  of  the  charge  —^f. 
Superpose  tlie  distribution  —  a-,  u[)on  the  distribution  o-,  so  tliat 
the  total  charge  sliall  be  equal  to  zero ;  then  tlie  surface  <liMisity 
at  every  point  is  n-i— o-o,  and  this  must  be  zero  by  wliat  we  liave 
just  proved,  so  that  o-.  =  o-,. 

Since  we  may  superpose  on  the  same  conductor  a  numlK-r  of 
distributions,  each  one  of  which  is  by  itself  in  equililirium.  it  is 


108  ELECTEOSTATICo. 

easy^  to  see  that  if  the  whole  quantity*  of  electricity  on  any  con- 
ductor be  changed  in  a  given  ratio,  the  density  at  each  point 
will  be  changed  in  the  same  ratio. 

56.  ^Tubes  of  Force  and  their  Properties.  We  have  seen  that 
a  unit  of  positive  electricit}'  concentrated  at  a  point  P  just  out- 
side a  conductor  would  be  urged  away  from  the  conductor  or 
drawn  towards  it,  according  as  that  point  on  the  conductor  which 
is  nearest  P  is  positively  or  negatively  electrified.  If  we  regard 
lines  of  force  drawn  in  an  electric  field  as  generated  by  points 
moving  from  places  of  higher  potential  to  places  of  lower  poten- 
tial, we  may  sa}-  that  a  line  of  force  2^^oceeds  from  every  point 
of  a  conductor  where  the  surface  density  is  positive,  and  that  a 
line  of  force  ends  at  every  point  of  a  conductor  where  the  sur- 
face density'  is  negative.  No  line  of  force  either  leaves  or 
enters  a  conductor  at  a  point  where  the  surface  density-  is  zero, 
and  no  line  of  force  can  start  at  one  point  of  a  conductor  where 
the  electrification  is  positive  and  return  to  the  same  conductor 
at  a  point  where  the  electrification  is  negative.  No  line  of  force 
can  proceed  from  one  conductor  at  a  point  electrified  in  anj-  way 
and  enter  another  conductor  at  a  point  where  the  electrification 
has  the  same  name  as  at  the  starting-point.  A  line  of  force 
never  cuts  through  a  conductor  so  as  to  come  out  at  the  other 
side,  for  the  force  at  every  point  inside  a  conductor  is  zero. 

Lines  and  tubes  of  force  are  sometimes  called  in  electrostatics 
lines  and  tubes  of  "  induction." 

"When  a  tube  of  force  joins  two  conductors,  the  charges  Qi, 
Q2  of  the  portions  JSi,  S^  which  it  cuts  from  the  two  surfaces  are 


Fig.  38. 


made  up  of  equal  quantities  of  opposite  kinds  of  electricity. 
For  if  we  suppose  the  tube  of  force  to  be  arbitrarily  prolonged 


ELECTROSTATICS.  109 

and  closed  at  the  ends  inside  tlie  two  conductors,  tlie  surface 
integral  of  normal  force  taken  over  the  box  thus  formed  is  zero, 
for  the  part  outside  the  conductors  yields  nothing,  since  the  re- 
sultant force  is  tangential  to  it,  and  there  is  no  resultant  force 
at  any  point  inside  a  conductor.  It  follows,  from  Gauss's 
Theorem,  that  the  whole  quantity  of  electricity  (Q,  -{-Qj)  inside 
the  box  must  be  zero,  or  Q,  =  —  Q.,,  which  proves  the  theorem. 
If  o-j  and  0-2  are  the  average  values  of  the  surface  densities  of 
the  charges  on  Si  and  S.,  respectively,  we  have  (riSi  =  Qi  and 
0-2-82  =  ^25  whence 

o-2  =  -rri^  [162] 

The  integral  taken  over  any  surface,  closed  or  not,  of  the 
force  normal  to  that  surface  is  called  by  some  writers  the  Jlow 
of  force  across  the  surface  in  question,  and  by  others  the  induc- 
tion through  this  surface. 

If  we  apply  Gauss's  Theorem  to  a  box  shut  iu  by  a  tube 
of  force  and  the  i)ortions  S^  S.,  which  it  cuts  from  any  two 
equipotential  surfaces,  we  shall  have,  if  the  box  contains  no 
electricity, 

F.S2-I\Si  =  0,  [163] 

where  Fi  and  F.^  are  the  average  values,  over  S^  and  So  respec- 
tively, of  the  normal  force  taken  iu  the  same  direction  (that  in 
which  T^  decreases)  in  both  cases.  In  other  words,  the  flow  of 
force  across  all  equipotential  sections  of  a  tube  of  force  con- 
taining no  electricity  is  the  same,  or  the  average  force  over  an 
equipotential  section  of  an  empty  tube  of  force  is  inversely  pro- 
portional to  the  ax'ea  of  the  section. 


Fir..  .",9. 


"When  a  tube  of  force  encounters  a  quantity  m  of  electricity 
(Fig.  39),  the  flow  of  force  through  the  tube  on  passing  this 


110  ELECTROSTATICS. 

electricity  is  increased  by  47rm.  If,  however,  the  tube  encoun- 
ters a  conductor  large  enough  to  close  its  end  completely,  a 
charge  ni  will  be  found  on  the  conductor  just  sufficient  to  reduce 
to  zero  the  flow  of  force  (7)  through  the  tube.     That  is, 

/ 

m  = -• 

It  is  sometimes  convenient  to  consider  an  electric  field  to  be 
divided  up  by  a  system  of  tubes  of  force,  so  chosen  that  the  flow 
of  force  across  any  equipotential  surface  of  each  tube  shall  be 
equal  to  47r.  Such  tubes  are  called  unit  tubes;  for  wherever 
one  of  them  abuts  on  a  conductor,  there  is  always  the  unit  quan- 
tity of  electricity  on  that  portion  of  the  conductor's  surface  which 
the  tube  intercepts.  In  some  treatises  on  electricity  tlie  term 
"line  of  force"  is  used  to  represent  a  unit  tube  of  force,  as 
when  a  conductor  is  said  to  cut  a  certain  number  of  "  lines  of 
force." 

It  is  evident  that  m  unit  tubes  abut  on  a  surface  just  outside 
a  conductor  charged  with  m  units  of  either  kind  of  electricity, 
if  the  superficial  density  of  the  charge  has  everywhere  the  same 
sign.  These  tubes  must  be  regarded  as  beginning  at  tlie  cou- 
ductor  if  m  is  positive,  and  as  ending  there  if  m  is  negative. 
If  a  conductor  is  charged  at  some  places  with  positive  elec- 
tricity and  at  others  with  negative  electricity,  tubes  of  force 
will  begin  where  the  electrification  is  positive,  and  others  will 
end  where  the  electrification  is  negative. 

It  is  evident  that  no  tube  of  force  can  return  into  itself. 


Fig.  40. 


57.   Hollow   Conductors.      When  the  nonconducting  cavity, 
shut  in  by  a  hollow  conductor  K  (Fig.  40),  contains  quantities 


ELECTROSTATICS.  Ill 

of  electricity  (mj,  wig,  ^3,  etc.,  or  ^   vi)  distributed  in  any  way, 

but  insulated  from  A",  there  is  induced  on  tlie  walls  of  the  cavity 
a  charge  of  electricity  algebraically  equal  in  quantity,  but  oi)po- 
site  in  sign,  to  the  algebraic  sum  of  the  electricity  within  the 
cavity. 

Call  the  outside  surface  of  the  conductor  S„  and  its  charge 
3/0,  the  boundary  of  the  cavity  *S',-  and  its  charge  J/,,  and  sur- 
round the  cavity  by  a  closed  surface  <6',  every  point  of  which  lies 
within  the  substance  of  the  conductor,  where  the  resultant  foice 
is  zero.  Now  the  surface  integral  of  normal  force  taken  over 
S  is  zero,  so  that,  according  to  Gauss's  Theorem,  the  algebraic 
sum  of  the  quantities  of  electricity  within  the  cavity  and  upon 
Si  is  zero.     That  is, 

3/.  +  ,„^  _|_ ^n,  +  mg  +  •••  =  .1/.  +  V  (/h)  =  0,        [1G4] 

and  this  is  our  theorem,  which  is  true  whatever  the  charge  on 
S„  is,  and  whatever  distribution  of  free  electricity  there  may 
be  outside  A".  If  the  distribution  of  the  electricity  within  the 
cavity  be  changed  by  moving  ?»i,  yn.j,  etc.,  to  ditTerent  positions, 
the  (lifitriJmtinn  of  3/,  on  *S',  will  in  general  be  changed,  although 
its  value?  remains  unchanged. 

If  K  has  received  no  electricity  from  without,  its  total  charge 
must  be  zero  ;  that  is, 

3/:,  =  -3/.='V(m). 

If  a  charge  algebraically  equal  to  M  be  given  to  A', 
3/„  =  3/-3/.. 

The  combined  effect  of  ^  (m ),  the  electricity  within  the  cavity, 

and  J/j,  the  electricity  on  the  walls  of  the  cavity,  is  at  all  points 
without  Si  absolutely  null.  For,  if  we  ajiply  [l-">;5]  toS,  any  sur- 
face drawn  in  the  conductor  so  as  to  enclose  >',.  we  shall  have  />,.! ' 
everywhere  zero,  since  the  poti'utial  function  is  constant  within 
the  conductor ;  this  shows  that  Vi,  the  potential  function  due  to 


112  ELECTKOSTATICS. 

all  the  electricity  within  S^  must  be  zero  at  all  points  without  S ; 
but  /S  may  be  drawn  as  nearly  coincident  wilh  aS,-  as  we  please. 
Hence  our  theorem,  which  shows  that,  so  far  as  the  value  of  the 
potential  function  in  the  substance  of  the  conductor  or  outside 
it,  and  so  far  as  the  arrangement  of  M„  and  of  3/',  any  free 

electricity  there  may  be  outside  K,  are  concerned,  Mj  and  ^    (?«,) 

might  be  removed  together  without  changing  anything.  The 
potential  function  at  all  points  outside  Si  is  to  be  found  by  con- 
sidering only  J/ and  M'. 

If  Si  happens  to  be  one  of  the  equipotential  surfaces  of  ^    (m) 

considered  by  itself.  Mi  will  be  arranged  in  the  same  way  as  a 
charge  of  the  same  magnitude  would  arrange  itself  on  a  con- 
ductor whose  outside  surface  was  of  the  shape  Si,  if  removed 
from  the  action  of  all  other  free  electricity. 

The  potential  function  ( V2)  due  to  M„  and  M'  is  constant 
everywhere  within  S„ ;  for  if  we  apply  [157]  to  a  surface  S, 
drawn  within  the  substance  of  the  conductor  as  near  Sg  as  we 

like,  we  shall  have 

F.-Fo  =  0, 

which  proves  the  theorem. 

The  potential  function  within  the  cavity  is  equal  to  V2  +  F"i, 

where  Vi  is  the  potential  function  due  to  3fi  and  ^   {m).  Of  these, 

V2  is,  as  we  have  seen,  constant  throughout  K  and  the  cavity 
(Section  31)  which  it  encloses,  while  Vi  has  different  values  in 
different  parts  of  the  cavity,  and  is  zero  within  the  substance  of 
the  conductor. 

Suppose  now  that,  by  means  of  an  electrical  machine,  some 
of  the  two  kinds  of  electricity  existing  combined  together  in  a 
conductor  within  the  cavity  be  separated,  and  equal  quantities 
(q)  of  each  kind  be  set  free  and  distributed  in  any  manner 
within  the  cavity. 

The  value  of  Vi  within  the  cavity  will  probably  be  different 
from  what  it  was  before,  but   V^  will  be  unchanged ;   for  the 


ELECTROSTATICS.  113 

quantity  of  matter  in  the  cavity  is  unchanged,  being  now,  alge- 
braically considered, 


^(m)  +g-9=2^(m), 


so  that  J/j  is  unchanged,  although  it  may  have  been  differently 
arranged  on  Si,  in  order  to  keep  the  value  of  Vi  zero  within 
the  substance  of  the  conductor.  If  now  a  part  of  the  free 
electricity  in  the  cavity  be  conveyed  to  S^  in  some  way,  the  sub- 
stance of  the  conductor  will  still  remain  at  the  same  potential  as 
before.  For,  if  I  uuits  of  positive  electricity  and  n  units  of 
negative  electricity  be  thus  transferred  to  S^,  the  whole  quantity 

of  free  electricity  within  the  cavit}'  will  be   y    (m)  —  Z  +  ?i,  and 

that  on  Si  will  be  3/^  +  ?  — ?i :  but  these  are  numerically  equal, 
but  opposite  in  sign,  and  the  charge  on  Si,  if  properly  arranged, 
suffices,  without  drawing  on  J/„  to  reduce  to  zero  the  value  of 
Vi  in  K.  Since  M„  and  M'  remain  as  before,  V.,  is  unchanged, 
and  the  conductor  is  at  the  same  potential  as  before.  So  long 
as  no  electricity'  is  introduced  into  the  cavity  from  loithoiit  K, 
no  electrical  charges  within  the  cavity  can  have  any  effect  out- 
side Si. 

Most  experiments  in  electricity  are  carried  on  in  rooms,  which 
we  can  regard  as  hollows  in  a  large  conductor,  the  earth.  V>i, 
the  value  of  the  potential  function  in  tlie  earth  and  the  walls  of 
the  room,  is  not  changed  by  anything  that  goes  on  inside  the 
room,  where  the  potential  function  is  F=  T^i  +  T'o.  Since  we 
are  generally  concerned,  not  with  the  absolute  value  of  the  poten- 
tial function,  but  only  with  its  variations  within  the  room,  and 
since  T^.  remains  always  constant,  it  is  often  convenient  to  dis- 
regard F2  altogether,  and  to  call  F,  the  value  of  the  potential 
function  inside  the  room.  When  we  do  this  we  must  remember 
that  we  are  taking  the  value  of  the  potential  function  in  the 
earth  as  an  arbitrary  zero,  and  that  the  v.aluo  of  T',  at  a  point  in 
the  room  really  measures  only  the  difTerence  between  the  values 
of  the  potential  function  in  the  earth  and  at  the  point  in  ques- 
tion.    AVhen  a  conductor  A  in  the  room  is  connected  with  the 


114  ELECTROSTATICS. 

walls  of  the  room  by  a  wire,  the  value  of  Vi  in  A  is,  of  course, 
zero,  and  A  is  said  to  have  been  put  to  earth. 

58.  Induced  Charge  on  a  Conductor  which  is  put  to  Earth. 

Suppose  that  there  are  in  a  room  a  number  of  conductors,  viz.  : 
Ai  charged  with  M^  units  of  electricity,  and  A2,  Aj,  A^,  etc., 
connected  with  the  walls  of  the  room,  and  therefore  at  the  po- 
tential of  the  eai'th,  which  we  will  take  for  our  zero.  If  the 
potential  function  has  the  value  pi  inside  Ai,  everj'  point  in  the 
room  outside  the  conductors  must  have  a  value  of  tlie  potential 
function  l^'ing  between  2h  ^^^^  0,  else  the  potential  function  must 
have  a  maximum  or  a  minimum  in  empty  space.  If  pi  is  posi- 
tive, there  can  be  no  positive  electricity  on  the  other  conductors  ; 
for  if  there  were,  lines  of  force  must  start  from  these  conductors 
and  go  to  places  of  lower  potential ;  but  there  are  no  such  places, 
since  these  conductors  are  at  potential  zero,  and  all  otlier  points 
of  the  room  at  positive  potentials.  In  a  similar  wa}'  we  may 
prove  that  if  2h  is  negative,  the  electricity  induced  on  the  other 
conductors  is  wholly  positive. 

Now  let  us  apply  [158]  to  a  spherical  surface,  drawn  so  as 
to  include  Ai  and  at  least  one  of  the  other  conductors,  but  with 
radius  a  so  small  that  some  parts  of  the  surface  shall  lie  within 
the  room.  If  we  take  the  point  0  at  the  centre  of  this  surface, 
we  shall  have 

47rF2  =  i  CDrV'ds  +  -^  fvds.  [165] 

etc/  Ct  \J 

If  M  is  the  whole  quantity  of  electricity  within  the  spherical 

surface,  there  must  be  a  quantity  —J!f  outside  the  surface,  either 

on  the  walls  of  the  room  or  on   conductors  within  the  room. 

The  value  at  0  of  the  potential  function,  F^?  due  to  the  elec- 

M 

tricity  without  the  sphere,  is  less  in  absolute  value  than , 

a 

for  it  could  only  be  as  great  as  this  if  all  the  electricity  outside 
the  sphere  were  brought  up  to  its  surface. 
By  Gauss's  Theorem, 

'AF.ds  =  -47ri¥, 


/^ 


ELECTROSTATICS.  115 

therefore,  Cvds  =  4 Tra  [3/+  a F,] .  [166] 

Now,  if  Ml  is  positive,  the  integral  is  positive,  for  all  parts  of 
the  spherical  surface  within  the  room  j'ield  positive  differentials, 
and  all  other  parts  zero,  so  that  the  second  side  of  the  equation 
is  positive.  But  a  Vo  is  of  opposite  sign  to  3/,  and  is  less  in 
absolute  value ;  hence,  M  is  positive,  and  the  total  amount  of 
negative  electricity  induced  on  the  other  conductors  within  the 
spherical  surface  by  the  charge  on  yl,,  is  numerically  less  than 
this  charge,  unless  some  one  of  these  conductors  suiTounds  ^1, ; 
in  which  case  the  induced  chargo  comes  wholly  on  this  conduc- 
tor, while  the  other  conductors,  and  the  walls  of  the  room,  are 
free.  Some  of  the  tubes  of  force  which  begin  at  A^  end  on  the 
walls  of  the  room,  provided  these  latter  can  be  reached  from 
Ai  without  passing  through  the  substance  of  an}-  conductor. 

59.  Coefficients  of  Induction  and  Capacity.  If  a  number  of 
insulated  conductors,  A2,  A-,,,  ^li,  etc.,  are  in  a  room  in  the  pres- 
ence of  a  conductor  A^  charged  with  Mi  units  of  electricity,  the 
whole  charge  on  each  is  zero  ;  but  equal  amounts  of  positive  and 
negative  electricity  are  so  arranged  by  induction  on  each,  that 
the  potential  function  is  constant-  throughout  the  substance  of 
every  one  of  the  conductors. 

Let  the  values  of  the  potential  functions  in  the  system  of  con- 
ductors be  />!,  2hi  Psi  Pii  etc.  Since  each  conductor  except  Ai  is 
electrified,  if  at  all,  in  some  places  with  positive  electricit}',  and 
in  others  with  negative  electricity,  some  lines  of  force  must 
start  from,  and  others  end  at,  every  sucii  electrified  conductor, 
so  that  there  must  be  points  in  the  air  about  each  conductor  at 
lower  and  at  higher  potentials  than  the  conductor  itself.  But 
the  value  of  the  potential  function  in  the  walls  of  the  room  is 
zero,  and  there  can  be  no  ])oints  of  maximum  or  minimum  poten- 
tial in  empty  space  ;  so  tliat  pi  must  be  that  value  of  the  poten- 
tial function  in  the  room  most  widely  different  from  zero,  and 
P2,  ^3,  Pi,  etc.,  must  have  the  same  sign  as  ;>,. 

The  reader  may  show,  if  he  likes,  that  both  the  negative  part 


116  electhostatics. 

and  the  positive  pai"t  of  the  zero  charge  of  any  conductor,  ex- 
cept Ai,  is  less  than  J/j. 

Letpu  be  the  vahie  of  the  potential  function  in  a  conductor 
A^  charged  with  a  single  unit  of  electricit3-,  and  standing  in 
the  presence  of  a  number  of  other  conductors  all  uncbarged 
and  insulated.  Then  if  Pui  Ihsi  Pui  etc.,  are,  under  these  cir- 
cumstances, the  values  of  the  potential  functions  in  the  other 
conductors.  A.,,  A-^,  A^,  etc.,  the  potential  functions  in  these 
conductors  will  be  M^p^zi  -M,pi3,  J/ij^u,  etc.,  if  ^li  be  charged 
with  Ml  units  of  electricit\-  instead  of  with  one  unit.  This  is 
evident,  for  we  may  superpose  a  number  of  distributions  Avhich 
are  singly  in  equilibrium  upon  a  set  of  conductors,  and  get  a 
new  distribution  in  equilibrium  where  the  density  is  the  sum  of 
the  densities  of  the  component  distributions,  and  the  value  of 
the  resulting  potential  function  the  sura  of  the  values  of  their 
potential  functions. 

If  Ai  be  discharged  and  insulated,  and  a  charge  JL  be  given 
to  Ao,  the  values  of  the  potential  functions  in  the  different  con- 
ductors may  be  written 

Mop.2i,  M,p22,  M.psa,  M^Pn,  etc. 

If  now  we  give  to  A^  and  Ao  at  the  same  time  the  charges  Jfj 
and  M.,  respectively,  and  keep  the  other  conductoi's  insulated, 
the  result  will  be  equivalent  to  superposing  the  second  distribu- 
tion, which  we  have  just  considered,  upon  the  first,  and  the  con- 
ductors will  be  respectively  at  potentials, 

Mi2'>n-\- ^J^-iP2i^   ^^iPi2  +  ^2Pz>,  MiPis  +  M2P2^,  etc.     [1G7] 

If  all  the  conductors  are  simultaneousl}'  charged  with  quanti- 
ties Jfj,  3/21  3fo,  3/4,  etc.,  of  electricity  respectively,  the  value 
of  the  potential  function  on  yl^  will  be 

V,  =  J/,Pu  +  3f,2),,  +  3/3P3,  +  ■...  +  M,p,,  +  3/,. 7),.,,  [108] 

Writing  this  in  the  form  V^  =  af^  + MkP„„,  we  see  that  if  the 
charges  on  all  the  conductors  except  ^^  be  unchanged,  a^will  be 

constant,  and  that  every  addition  of  —  units  of  electricitv  to 

Pa 


ELECTROSTATICS.  117 

the  charge  of  A^  raises  the  vahie  of  the  potential  function  in 
it  by  unity.  If  we  solve  the  n  equations  like  [168]  for  the 
charges,  we  shall  get  n  equations  of  the  form 

^V,  =  F,  5i,+  V,  q,,  4-  F3  ^3,  +  ...  +  F,ry,,  +  •••  +  F„  ?„„  [169] 
where  the  5's  are  functions  of  the  />'s. 

If  all  the  conductors  except  ^^  are  connected  with  the  earth, 
3/4  =  Vk  fjuc,  and  Q-ji  is  evidently  the  charge  which,  under  these 
circumstances,  must  be  given  to  A^  in  order  to  raise  the  vahie 
of  the  potential  function  in  it  by  unity.    It  is  to  be  noticed  that 

q^.  and  —  are  in  general  different. 

The  charge  ^hich  must  be  given  to  a  conductor  when  all  the 
conductors  which  surround  it  are  in  communication  with  the 
earth,  in  order  to  raise  the  value  of  the  potential  function  with- 
in that  conductor  from  zero  to  unity,  shall  be  called  the 
capacity  of  the  conductor.  It  is  evident  that  the  capacity  of  a 
conductor  thus  defined  depends  upon  its  shape  and  upon  the 
shape  and  position  of  the  conductors  in  its  neighborhood. 

60.  Distribution  of  Electricity  on  a  Spherical  Conductor. 
Considerations  of  symmetry  show  that  if  a  charge  3/  he  given 
to  a  conducting  sphere  of  radius  r,  removed  from  the  influence 
of  all  electricity  except  its  own,  the  charge  will  arrange  itself 
uniformly  over  the  surface,  so  that  the  superficial  density  shall 

be  evervwhere  (r  =  — — • 

The  value,  at  the  centre  of  the  sphere,  of  the  potential  function 

due  to  the  charge  3/ on  the  surface  is  1-,  and,  since  the  potential 

/■ 

function  is  constant  inside  a  charged  conductor,  this  must  be 

the  value  of  the  potential  function  throughout  the  sphere.     If  3/ 

is  equal  to  r,  -—  =  1  ;  hence  the  capacitv  of  a  spherical  conductor 
r 

removed  from  the  influence  of  all  electricity  except  its  own.  is 

numerically  equal  to  the  radius  of  its  surface. 


118  ELECTROSTATICS.    ' 

61.  Distribution  of  a  Given  Charge  on  an  Ellipsoid.     It  is 

evident  from  the  discussion  of  bomoeoids  in  Chapter  1.  that  a 
charge  of  electricity  arranged  (on  a  conductor)  in  the  form  of 
a  shell,  bounded  by  ellipsoidal  surfaces  similar  to  each  other 
(and  to  the  surface  of  the  conductor),  and  similarly  placed, 
would  be  in  equihbrium  if  the  conductor  were  removed  from  the 
action  of  all  electricity  except  its  own.  We  may  use  this  prin- 
ciple to  help  us  to  find  the  distribution  of  a  given  charge  on  a 
conducting  ellipsoid. 

Let  us  consider  a  shell  of  homogeneous  matter  bounded  by 
two  similar,  similarly  placed,  and  concentric  ellipsoidal  surfaces, 
whose  semi-axes  shall  be  respectively  a,  &,  c,  and  (l+a)a, 
(1  -\-a)b,  (1  +a)c.  If  any  line  be  drawn  from  the  centre  of 
the  shell  so  as  to  cut  both  surfaces,  the  tangent  planes  to  these 
two  surfaces  at  the  points  of  intersection  will  be  parallel,  and 
the  distance  between  the  planes  is  pa,  where  p  is  the  length 
of  the  perpendicular  let  fall  from  the  centre  upon  the  nearer  of 
the  planes. 

If  p  is  the  volume  densit}'  of  the  matter  of  which  the  shell  is 
composed,  the  mass  of  the  shell  is  Jf  =|^7ra&c  [(1  + a)''  — l]p, 
and  the  rate  at  which  the  matter  is  spread  upon  the  unit  of  sur- 
face is,  at  any  point,  cr  =  p8,  where  8  is  the  thickness  of  the 
shell  measured  on  the  line  of  force  which  passes  through  the 
point  in  question.    Eliminating  p  from  these  equations,  we  have 

o- = 11  ^0  I 

iTrabcla+a'  +  ia'^  ^         -^ 

If,  now,  in  accordance  with  the  hypothesis  that  the  thickness  of 
the  electric  charge  on  a  conductor  is  inappreciable,  we  make  a 
smaller  and  smaller,  noticing  that  8  differs  from  pa  by  an  infini- 
tesimal of  an  order  higher  than  the  first,  we  shall  have  for  a 
strictly  surface  distribution, 

cr  =  -^-.  [171] 

i-n-abc 

If  the  equation  of  the  surface  of  the  ellipsoidal  conductor  is 

^  4-  2/'  4_  ^'^  _  1 
a'      ¥       (f 


ELECTROSTATICS.  119 

we  have 


and 


1=     K 


c  _     I  •>  (  ^    i_lf 


a-       ¥ 


This  last  expression  shows  that,  as  c  is  made  smaller  and 
smaller,  o-  approaches  more  and  more  nearly  the  value 

M 

[172] 


Airah 


Ji_4_i:' 

\        a-'       6- 


and  this  gives  some  idea  of  the  distribution  on  a  thin  elliptical 
plate  whose  semi-axes  are  a  and  h. 

For  a  circular  plate,  we  may  put  a  =  6  in  the  last  expression, 
which  gives 

M 


ATva^/n- 


[173] 

for  the  surface  density  at  a  point  r  units  distant  from  the  centre 
of  the  plate. 

The  charge  3/ distributed  according  to  this  law  on  both  sides 
of  a  circular  plate  of  radius  a  raises  the  plate  to  potential 


a  Jo    ^/(j-'_  ,-'       2  a 


so  that'the  capacity  of  the  plate  is 

2a 


[174] 


62.  Spherical  Condensers.  If  a  conducting  sphere  A  of  radius 
r  (Fig.  41)  be  surrounded  by  a  concentric  splierical  conducting 
shell  B  of  radii  r,  and  r„  and  charged  with  m  units  of  electricity 
while  B  is  uncharged  and  insulated,  we  shall  have 

(1)  the  charge  m  uniformly  distributed  upon  aS,  the  surface 
of  the  sphere ; 

(2)  an  induced  charge  —  m  (Section  57)  uniformly  distributed 
upon  aSj,  the  inner  surface  of  B ; 


120 


ELECTROSTATICS. 


(3)    a  charge  +m  (since  the  total  charge  of  B  is  zero)  uni- 
formly distributed  on  S^^  the  outer  surface  of  B. 


Fig.  41. 


The  value  at  the  centre  of  the  sphere  of  the  potential  function 

nyy  aij  /yy^ 

due  to  all  these  distributions  is  F^  = '-  -\ ,  and  this  is 

the  value  of  V  throughout  the  conducting  sphere.     The  value  of 

771 

the  potential  function  in  B  is  V^  =  ~^- 

'  o 

If  now  a  charge  M  be  communicated  to  B,  this  will  add  itself 
to  the  charge  m  already  existing  on  S„,  and  the  charge  on  Si  will 
be  undisturbed.  The  values  of  the  potential  functions  in  the 
conductors  are  now 

__       m       m      m-\-  M         .    _..       m-\-  M 
V^  = 1 — — ,    and    V^  =  —~^ 

If  now  B  be  connected  with  the  earth  so  as  to  make  Fg  =  0, 
the  charges  on  S  and  Si  will  be  undisturbed,  but  the  charge  on 

So  will  disappear.      V^  is  now  equal  to 

r       Ti 

If  A  were  uncharged,  and  B  had  the  charge  M^  this  charge 
would  be  uniformly  distributed  upon  S^^  for,  since  the  whole 
charge  on  S  is  zero,  the  whole  cliarge  on  Si  must  be  zero  also. 
It  is  easy  to  see  that  S  and  ^^iraust  both  be  in  a  state  of  nature, 
for  if  not,  lines  of  force  must  start  from  S  and  end  at  >S„  and 
others  start  at  Si  and  end  at  S,  which  is  absurd. 


ELECTROSTATICS.  121 

If  A  were  put  to  earth  by  means  of  a  fine  insulated  wire 
passing  througli  a  tiny  liole  in  B,  and  if  B  were  insulated  and 
charged  with  M  units  of  electricity,  we  should  have  a  charge  x 
on  S,  a  charge  —  x  on  Si,  and  a  charge  M-\-x  on  S^.     To  find 

'V        OC         ^  A/" 

X,  we  need  only  remember  that  V^  =  ' 1 1 =  0,  whence 

X  may  be  obtained.  *       "        " 

If  B  be  put  to  earth,  and  A  be  connected  by  means  of  the  fine 
wire  just  mentioned,  with  an  electrical  machine  which  keeps  its 
prime  conductor  constantly  at  potential  Vi,  A  will  receive  a  charge 
y  and  will  be  put  at  potential  F^.  To  find  ?/,  it  is  to  be  noticed 
that  there  is  a  charge  —  y  on  -&',,  and  no  charge  on  S„,  which  is 

y       ?/ 
put  to  earth.      F^  =  ^  —  ^^  =  T^i,  whence  y  may  be  obtained. 

If  r  =  99  millimeters  and  r,=  100  millimeters,  y  =  9900  Vf 

If  a  sphere,  equal  in  size  to  A  but  having  no  shell  about  it, 

were  connected  with  the  same  prime  conductor,  it  too  would 

receive  a  charge  z  sufficient  to  raise  it  to  potential  Fj,  and  z 

would  be  determined  by  the  equation  Fi=  -•    If  r  =  99.  we  have 

r 
z  =  99  Vi ;    hence  we  see   that  A,  when   surrounded  by  B  at 

l)otential  zero,   is  able  to  take  one  hundred  times  as  great  a 

charge  from  a  given  machine  as  it  could  take  if  B  were  removed. 

In  other  words,    B  increases  ^I's  capacity  one   hundred   fold. 

A  and  B  together  constitute  what  is  called  a  condenser. 


Fig.  42. 


If  A  of  the  condenser  AB,  both  parts  of  which  are  supposed 
uncharged,  be  connected  by  a  fine  wire  (Fig.  42)  with  a  spliere 


122  ,  ELECTROSTATICS. 

A'  which  has  the  same  radius  as  A,  and  is  charged  to  potential 
Vi,  A  and  A'  will  now  be  at  the  same  potential  [F2],  and  A  will 
have  the  charge  x,  and  A'  the  charge  y.  The  total  quantity  of 
electricity  on  A'  at  first  was  rVi,  so  that  x  -\-y  =rVi,  and 

y      X      X       X 
I  ^      r      r       Ti      r„ 

whence  x  and  y  may  be  found. 

The  reader  may  study  for  himself  the  electrical  condition  of 
the  different  parts  of  two  equal  spherical  condensers  (Fig.  43) , 


Fig.  43. 

of  which  the  outer  surface  S^  of  one  is  connected  with  an  elec- 
tric machine  at  potential  Fi,  and  the  inside  of  the  other,  S\  is 
connected  with  the  earth.  The  two  condensers,  which  are  sup- 
posed to  be  so  far  apart  as  to  be  removed  from  each  other's 
influence,  illustrate  the  case  of  two  Leyden  jars  arranged  in 
cascade. 

63.  Condensers  made  of  Two  Parallel  Conducting  Plates. 
Suppose  two  infinite  conducting  planes  A  and  B  to  be  parallel 
to  each  other  at  a  distance  a  apart ;  choose  a  point  of  the 
plane  A  for  origin,  and  take  the  axis  of  x  perpendicular  to  the 
planes,  so  that  their  equations  shall  be  cc  =  0  and  x  =  a.  Let  the 
planes  be  charged  and  kept  at  potentials  V^  and  Vb  respectively. 
It  is  evident  from  considerations  of  symmetry  that  the  potential 
function  at  the  point  P  between  the  two  planes  depends  only 
upon  P's  X  coordinate,  so  that 

D„F=0,   i),F=0,     i>/F=0,   DJ'V=0. 


ELECTROSTATICS.  123 

Laplace's  Equation  gives,  then, 

whence  D^V=G,   and    V=Cx  +  D. 

If  ic  =  0,  V=  Vj, ;  and  if  a;  =  a,  F=  V^ ;  so  that 

V=(Vs-V,)l-]-V,,   and   Z),F=i^^. 

The  lines  of  force  are  parallel  between  the  planes,  and  the 
surface  densities  of  the  charges  on  A  and  B  are 

— and  — ~ respectively. 

47ra  A-Tra 

If  we  take  a  portion  of  area  S  out  of  the  middle  of  each  plate, 

there  will  be  a  quantity  of  electricity  on  S^  equal  to     ^     ' -~^', 

and  an  equal  quantity  of  the  other  kind  of  electricity  on  S^. 
The  force  of  attraction  between  -S,,  and  iS^  will  be  2Tra'''S,  or 

Hit  Cv" 

If  Sn  be  put  to  earth,  the  charge  that  must  be  given  to  S^  in 
order  to  raise  it  to  potential  unity  is 

Aira 
In  other  words,  the  capacity  of  S^  is  inversely  proportional  to 
the  distance  between  the  plates. 

In  the  case  of  two  thin  conducting  plates  placed  parallel  to  and 
opposite  each  other,  at  a  distance  small  compared  with  their 
areas,  the  lines  of  force  are  practically  parallel  except  in  the 
immediate  vicinity  of  the  edges  of  the  plates  ;*  and  we  may  infer 


Vb 
Fig.  44. 


*  See  Maxwell's  Treatise  on  Electricity  and  Maynttism,  Vol.  I.  Fig.  XII. 


124  ELECTROSTATICS. 

from  the  results  of  this  section  that  the  capacity  of  a  condenser 
consisting  of  two  parallel  conducting  plates  of  area  S,  separated 
by  a  layer  of  air  of  thickness  a,  when  one  of  its  plates  is  put  to 

earth  is  very  approximately for  large  values  of  -  • 

Aira  a 

64.  Capacity  of  a  Long  Cylinder  surrounded  by  a  Concentric 
Cylindrical  Shell.  In  the  case  of  an  infinite,  conducting  cylinder 
of  radius  r^,  kept  at  potential  F;  and  surrounded  by  a  concentric 
conducting  cylindrical  shell  of  radii  r„  and  r',  kept  at  potential 
Vo,  we  have  symmetry  about  the  axis  of  the  cylinder,  so  that 
D^  V—  0,  and  Laplace's  Equation  reduces  to  the  form 

whence,  for  all  points  of  empty  space  between  the  cylinder  and 

its  shell,  T7-      n  ,   T\A 

'  V=  C  -\-D\ogr. 

But   V=  Vi  when  r  =  ri,  and  F=  Vo  when  r=  r^, 

FJog^^  +  FJog^ 

hence  F= '■>  [1^3] 

log^° 

and  nF=i^»-^^- 


To      r 


Fig.  45. 


The  surface  densities  of  the  electricity  on  the  outer  surface 
of  the  cylinder  and  the  inner  surface  of  the  shell  are  respectively 


ELECTllOSTATICS.  12^ 

— I —     aiul     2 L., 


4  Trr,  log  _?  4  n-r„  log 


so   that  the  charge   on  the   unit  of  length  of  the   cylinder  is 

V  —  V 

— : 2,   and  the  charge  on  the  corresponding  portion   of  the 

21og:." 

inner  surface  of  the  shell  is  the  negative  of  this.     "We  may  find 
the  capacity  of  the  unit  length  of  the  cylinder  by  putting  F„  =  0 

and  Vi=  I,  whence  capacity  =  - 


2  loir !} 


If  r„  in  tins  expression  is  made  very  large,  the  capacity  of  the 
cylinder  will  be  very  small. 

In  the  case  of  a  fine  wire  connecting  two  conductors,  r^  will 
be  very  small,  and  there  will  be  no  conducting  shell  nearer  than 
the  walls  of  the  room,  so  that  the  capacity*  of  such  a  wire  is 
plainly  negligible. 

65.  Specific  Inductive  Capacity.  In  all  our  worlc  up  to  this 
time  we  have  supposed  conductors  to  be  separated  from  each 
other  by  electrically  indifferent  media,  which  simply  prevent 
the  passage  of  electricity  from  one  conductor  to  another.  We 
have  no  reason  to  believe,  however,  that  such  media  exist. in 
nature.  Experiment  shows,  for  instance,  that  the  capacity  of 
a  given  spherical  condenser  depends  essentially  upon  the  kind 
of  insulating  material  used  to  separate  the  si)heie  from  its 
shell,  so  that  this  material,  without  conducting  elet'tririty. 
modifies  the  action  of  the  charges  on  the  conductoi's.  Insu- 
lators, when  considered  as  transmitting  electric  action,  arc 
sometimes  called  dielectrics. 

Whatever  may  really  be  the  physical  natures  of  the  sub- 
stances, such  as  glass,  parafline,  el)onite.  etc.,  which  we  com- 
monly use  as  insulators,  it  has  been  shown  that  their  behavior 
would  be  fairly  well  accounted  for  on  the  supi)osition  that  they 


126  ELECTEOSTATICS. 

are  made  up  of  truly  insulating  matter  in  which  are  imbedded, 
at  little  distances  from  one  another,  small,  conducting  par- 
ticles. It  is  evident  that  every  such  particle,  if  lying  in  a 
field  of  force,  would  be  polarized  ;  that  is,  one  part  would  be 
charged  positively  by  induction,  and  the  part  most  remote 
from  this  would  be  charged  negatively,  and  that  these  induced 
charges  would  have  some  influence  in  determining  the  A'alues 
of  the  potential  function  at  points  in  the  dielectric  and  in  the 
conductors  adjacent  to  it. 

Using  the  notation  of  Section  62,  let  the  part  A  of  a  spheri- 
cal condenser  be  charged  with  m  units  of  positive  electricity 
and  separated  from  the  part  B,  which  is  put  to  earth,  by  a 
spherical  shell  of  radii  r  and  r^  made  up  of  a  given  dielectric. 
Let  us  first  ask  ourselves  what  the  effect  of  the  dielectric  would 
be  if  it  consisted  of  extremely  thin  concentric  conducting  spheri- 
cal shells  separated  by  extremely  thin  insulating  spaces.  It  is 
evident  that  in  this  case  we  should  have  a  quantity  —m  of  elec- 
tricity induced  on  the  inside  of  the  innermost  shell,  a  quantity' 
-\-m  on  the  outside  of  this  shell,  a  quantity  — m  on  the  inner 
surface  of  th6  next  shell,  a  quantity  +m  on  the  outside  of  this 
shell,  and  so  on.  If  there  were  n  such  shells  in  the  dielectric 
layer,  and  ?t  +  1  spaces,  and  if  S  were  the  distance  from  the 
inner  surface  of  one  shell  to  the  inner  surface  of  the  next, 
and  XS  the  thickness  of  each  shell,  the  value,  at  the  centre  of 
A,  of  the  potential  function  due  to  the  charges  on  these  shells, 
would  be 


F. 


1  1      _  >       1  1 


r  +  8      r— A5  +  S      r  +  2S      r  — AS  +  28 

r  1 


+  •••  + 


r  4-  n8      r  —  AS  +  nS 
r  1 

=  —  wi  AS 


'         +,  .....'  ^..„..+^ 


_{r+8)  (r-AS+8)       (r+2S)  (r-A3  +  2  S) 
This  quantity  lies  between 


*=?i 


G  =  —  m\8y   :;  and  II=:  —  mX8/ 

L^ir  +  JcBy  ^Ar  +  iay 


ELECTROSTATICS.  127 

but  these  differ  from  each  other  by  loss  than  e  =  m\8  ^'  ~— ,  so 

J  "^r  .-(,1-^)5  (J^  ''^I'i 

— ,    which    is   easily  seen  to  lie  between 
r  ar 

G  and  H^  differs  from  F^'  by  less  than  t.  If,  then,  8  is  very 
small  in  comparison  with  r  and  r,,  V2  differs  from  mxi ) 

by  an  exceedingh*  small  fraction  of  its  own  value. 

This  shows  that  the  effect,  at  the  centre  of  ^-1,  of  such  a 
system  of  conducting  shells  as  we  have  imagined  would  be 
practically  the  same  as  if  a  charge  —m\  were  given  to  the 
inner  surface  of  the  dielectric,  and  a  charge  -j-mA  to  its  outer 
surface,  while  the  charges  on  the  surfaces  of  the  thin  shells 
within  the  mass  of  the  dielectric  were  taken  away.  That  is, 
the  value  of  the  potential  function  in  ^1  would  be 

m(l— A)( )   instead  of  m  { 

Such  a  system  of  shells  introduced  into  what  we  have  hitherto 
supposed  to  be  the  electrically  inert  insulating  matter  between 
the  two  parts  of  a  spherical  condenser  would  increase  the  capa- 
city of  the  condenser  in  the  ratio  of  1  to  1  —  A.  It  is  to  be 
noticed  that  A  is  a  proper  fraction  :  A  =  0  and  A  =  1  would 
correspond  respectively  to  a  perfect  insulator  and  to  a  perfect 
conductor. 

As  Dr.  E.  II.  Hall  has  suggested  to  me,  tlie  result  given 
above  might  be  easily  obtained  b}'  computing*  the  amount  of 
work  done  in  moving  a  unit  particle  of  ek-ctricity  (.supposed 
to  be  concentrated  at  a  point,  and  not  to  disturl)  existing  dis- 
tributions) from  A  to  B.  It  is  easy  to  see  that  the  force  at 
anj'  point  in  the  mass  of  one  of  the  thin  conducting  shells 
would  be  zero,  and  that  the  force  at  any  point  in  tlie  space 
between  two  shells  would  be  exactly  the  same  as  if  there  were 
no  shells  in  the  dielectric.  "We  have  no  reason  to  think  that 
there  are  any  such  differences  between  the  values  of  the  force 
at  contiguous  points  in  the  dielectric  as  this  would  indicate, 
and  the  conception  of  the  tliin  shells  has  been  introduced  only 

*  Mascart  et  Joubert,  Le<;ons  sur  I' Electricity,  §  124. 


128  ELECTROSTATICS. 

because  the  effect  of  these  shells  can  be  more  easily  computed 
thau  that  of  a  number  of  discrete  particles. 

When,  however,  the  dielectric  between  the  parts  of  a  spheri- 
cal condenser  is  supposed  to  contain  not  a  system  of  continuous 
shells,  but  a  number  of  separate  conducting  particles,  these  are 
often  regarded  as  forming  a  series  of  concentric  layers,  and  it  is 
assumed  that  the  sum  of  the  charges  induced  on  the  inner 
sides  of  the  particles  in  the  innermost  layer  is  —  X'm,  where 
X'  is  a  proper  fraction,  larger  or  smaller  in  different  dielectrics 
according  as  the  particles  are  nearer  together  or  farther  apart, 
and  that  the  inner  surfaces  of  all  the  other  layers  have  each 
the  same  charge,  and  the  outer  surface  of  every  layer  the  cor- 
responding positive  charge  +  A'm.  The  effect  of  this  kind  of 
dielectric,  if  made  to  replace  a  perfect  insulator  in  our  calcu- 
lations, would  be  to  increase  the  capacity  of  the  condenser  in 
the  ratio  1  to  1  —  /x,  where  fi  =  X'X,  and  it  is  evident  that  the 
same  effect  might  be  produced  by  a  charge  —  jxm  on  that 
surface  of  the  dielectric  which  touches  A^  and  a  charge  -j-  fxm 
on  that  surface  which  is  in  contact  witli  B. 

Experiment  shows  that  dielectrics  used  to  separate  and  to 
surround  charged  conductors  behave,  in  many  respects,  as  if 
every  surface  in  contact  with  a  conductor  had  a  charge  opposite 
in  sign  to  that  of  the  conductor,  and  in  absolute  value  /x  times 
as  great,  [x  being  less  than  unity,  and  constant  for  any  one 
dielectric.  That  is,  if  the  dielectric  separating  from  each  other 
a  number  of  conductors  be  displaced  by  another,  the  capacities 
of  all  the  conductors  will  be  changed  in  the  same  ratio,  depend- 
ing only  upon  the  natures  of  the  two  dielectrics. 

The  ratio  of  the  fraction ,  in  the  case  of  any  dielectric  to 

the  same  fraction  in  the  case  of  'air,  for  which  /x  is  very 
nearly  the  same  as  for  what  we  call  a  vacuum,  is  called  the 
S2)ecijlc  inductive  capacity  of  the  dielectric  in  question.  This 
ratio  is  greater  than  unity  for  all  solid  and  liquid  dielectrics 
with  which  we  are  acquainted.  The  specific  inductive  capacity 
of  a  perfect  conductor  would  be  infinite. 


ELECTROSTATICS.  129 

The  following  very  clear  statement  of  the  effect  produced  by 
changing  the  dielectric  which  envelops  the  parts  of  a  condenser 
made  of  two  plates,  is  due  to  Dr.  Hall,  and  is  copied  with  his 
permission : 

"  The  fundamental  fact  concerning  static  electrical  induction 
as  observed  b}'  Faraday  is  this,*  that  if  the  two  plates  of  a 
condenser,  separated  by  air,  receive  respectiveh"  e^  and  —  e.2 
units  of  electricity  when  charged  to  a  certain  difference  of 
potential,  e.g.,  by  connection  with  the  poles  of  a  battery  of 
man}'  cells  in  series,  the  same  two  plates  would,  if  any  other 
medium  were  substituted  for  the  air,  other  conditions  remaining 
unchanged,  receive  respectively  7v>i  and  —  Ke.j  units  of  electric- 
ity, K  being  some  quantity  greater  than  unity.  This  quantity 
iris  called  the  specific  inductive  capacity  of  the  second  medium. 

"  Now,  since  the  difference  of  potential  between  A  and  B  is 
the  same  in  these  two  cases,  the  'electromotive  intensity,* f  i.e., 
the  force  exerted  upon  unit  quantity  of  electricity,  is  the  same 
in  the  two  cases  at  any  given  point  lying  in  the  region  through 
which  the  change  of  dielectric  extends.  If  we  were  to  attempt 
to  determine  the  surface  densities  of  the  charges  of  the  conduc- 
tors by  means  of  the  equation  J 

dv  dy 

the  values  obtained  would  be  the  same  for  both  cases.  These 
would  be  the  actual  values  of  the  surface  densities  if  air  were 
used,  but  would  evidently  not  be  the  actual  surface  densities 
for  the  other  case.  For  this  latter  case,  tlie  values  thus  found 
are  called  the  ^apparent'  surface  densities,  and  bear  to  the 
true  densities  the  ratio  1  to  K. 

""We  must  not  conclude  from  this  that  A  and  B  with  charges 
Kci  and  —  Ke.,  respectively  in  the  second  medium  would  act, 

*  ^faxweH's  Treatise  on  Klertrln'ti/  and  Mnfjneti.<m,  Art.  52. 
+  ^^a.\^vol^s  Treatise  on  Efftririti/  and  ^farJnelism,  Art.  44. 
J  ^[axwcll's  Treatise  on  E/ertriciti/  and  Magnetism,  First  Edition,  Art.  83. 
See,  also,  Section  47  of  tliis  book. 


130  ELECTROSTATICS. 

in  all  electrical  respects,  like  the  same  bodies  with  charges  Bi 
and  —  e.2  in  air.  Two  spheres,  A  and  i?,  in  air,  with  centres  at 
distance  r  from  each  other,  and  having   charges   e^   and  —e^ 

respectively,  would  attract  each  other  with  a  force  -i-^,  whereas 

the  same  two  spheres  with  actual  charges  Kbi  and  —  Ke^  in  a 
medium  of   specific   inductive  capacity  K  would  attract  each 

other  with  a  force*  — l-^.      This  seems  at  first  inconsistent 

r 

with  the  fact  that  the  electromotive  intensity  at  any  point,  as 
stated  above,  is  the  same  in  both  cases.  The  electromotive 
intensity  at  any  point,  however,  meaus  the  force  that  would  be 
exerted  upon  unit  actual  quantity  of  electricity  at  that  point, 
not  the  force  that  would  be  exerted  upon  unit  apparent  quan- 
tity.     S6  the  average  force  exerted  by  A's,  charge  upon  B's 

charge  in  either  of  our  two  cases  is  —^  for  each  actual  unit  of 

IT 

B's  charge.  Hence,  the  total  force  exerted  by  A  upon  B  is 
-i-?  for  the  first  case,  and  — -J-^  for  the  second  case,  as  stated 
before." 

66.  Charge  induced  on  a  Sphere  by  a  Charge  at  an  Outside 
Point.  The  value  at  any  point  P  of  the  potential  function  due 
to  nil  units  of  positive  electricity  conceutrated  at  a  point  A^,  and 
m2  units  of  negative  electricity  concentrated  at  a  point  -dg,.  is 

mi      ma 
F  = where     ?'i  =  A^P  and  r^  =  A2  P. 

It  is  easy  to  see  that  if  mj  is  greater  than  ??i2i  so  that  mi=  Amg 
where  A  >  1 ,  F  will  be  equal  to  zero  all  over  a  certain  sphere 
which  surrounds  A^. 

If  (Fig.  4G)  we  let  AiA,,=  a,  AiO  =  8^,  A^O^S^,  OD  =  r, 
it  is  eas}-  to  see  that 

A-  —  1  K-—  1  {X.^  —  1)^  62 

*  Maxwell's  Treatise  on  Electricity  and  Magnetism,  Art.  94. 


ELECTROSTATICS.  131 

and  a  =  ^-^^-'^.  [;i76] 

If  PR  represents  the  force  /i  due  to  the  electricity  at  A^,  and 
PQ  the  force  /,  due  to  the  electricity  at  A.,  the  line  of  action  of 
the  resultant  force  F  (represented  by  PL)  must  pass  through 
the  centre  of  the  sphere,  since  the  surface  of  the  sphere  is  equi- 
potential. 


Fig.  46. 

The  triangles  AiPO  and  A2PO  are  mutually  equiangular,  for 
they  have  a  common  angle  AyOP,  and  the  sides  including  that 
angle  are  proportional  (1^  =  6182).  Hence,  from  the  triangles 
QPL  and  A^PAo,  by  the  Theorem  of  Sines, 

F 


A 

A 

sin  tti 

sinuj 

n 

r.2 

sin(a2 


.    ,  -.  [178] 

smoo      smai      sni  (tto  —  aiy 

whence  F  =  -^^  = 5.  =  . — J.  [1  <  9] 

Now,  according  to  Section  oO,  we  ma}-  distribute  upon  the 
spherical  surface  just  considered  a  quantity  vu  of  negative  elec- 
tricity in  such  a  way  that  the  effect  of  this  distribution  at  all 
points  outside  tlie  sphere  shall  be  equal  to  the  effect  of  the 
charge  —  ???2  concentrated  at  A.,,  and  the  effect  at  points  within 
the  sphere  shall  be  equal  and  opposite  to  the  effect  of  the  charge 
wix  concentrated  at  Ai.     Since  F  is  the  force  at  P  m  the  direc- 


132  ELECTROSTATICS. 

tion  of  the  interior  normal  to  the  spliere,  we  shall  accomplish 
this  if  we  make  the  surface  density  at  every  point  equal  to  cr, 
where 

and  if  we  now  take  away  the  charge  at  A^^  the  value  of  the  po- 
tential function  throughout  the  space  enclosed  by  our  spherical 
surface,  and  upon  the  surface  itself,  will  be  zero.  If  the  spheri- 
cal surface  were  made  conducting,  and  were  connected  with  the 
earth  by  a  fine  wire,  there  would  be  no  change  in  the  charge  of 
the  sphere,  and  we  have  discovered  the  amount  and  the  distri- 
bution of  the  electricity  induced  upon  a  sphere  of  radius  r,  con- 
nected with  the  earth  by  a  fine  wire  and  exposed  to  the  action 
of  a  charge  of  mx  units  of  positive  electricity  concentrated  at  a 
point  at  a  distance  S^  from   the  centime  of  tlie  sphere. 

If  now  we  break  the  connection  with  the  earth,  and  distribute 
a  charge  m  uniformly  over  the  sphere  in  addition  to  the  present 
distribution,  the  potential  function  will  be  constant  (although 
no  longer  zero)  within  the  sphere,  and  we  have  a  case  of  equi- 
librium, for  we  have  superposed  one  case  of  equilibrium  (where 
there  is  a  uniform  charge  on  the  sphere  and  none  at  A-^  upon 
another.     The  whole  chax'ge  on  the  sphere  is  now 

M  =  m  —  mo  =  m —, 

8, 

and  the  value  of  the  potential  function  within  it  and  upon  the 
surface, 

r        8i       r 

If  the  conducting  sphere  were  at  the  beginning  insulated  and 
uncharged,  we  should  have  3f=0,  and  therefore 

^  =  JBl(l--^'-'\    and    F=^.  [181] 

If  we  have  given  that  the  conducting  sphere,  under  the  influ- 
ence of  the  electricity  concentrated  at  Ai  is  at  potential  Vi,  we 


ELECTROSTATICS.  133 


know  that  its  total  charge  must  be  Fj  r  —  ^^ ,  and  its  surface 
density 

It  is  easy  to  see  that  the  sphere  and  its  charge  will  be  at- 
tracted toward  Ai  with  the  force 


and  the  student  should  notice  that,  under  certain  circumstances, 
this  expression  will  be  negative  and  the  force  repulsive. 

If  iHi  =  ??i2i  the  surface  of  zero  potential  is  an  infinite  plane, 
and  our  equations  give  us  the  charge  induced  on  a  conducting 
plane  by  a  charge  at  a  point  outside  the  i)lane. 

The  method  of  this  section  enables  us  to  find  also  the  capacity 
of  a  condenser  composed  of  two  conducting  cylindrical  siufuces, 
parallel  to  each  other,  but  eccentric ;  for  a  whole  set  of  the 
equipotential  surfaces  due  to  two  parallel,  infinite  straight  lines, 
charged  uniformly  with  equal  quantities  per  unit  of  length  of 
opposite  kinds  of  electricity,  are  eccentric  cylindrical  surfaces 
surrounding  one  of  the  lines  ^1.,  and  leaving  the  other  line  Ai 
outside.  AVe  may  therefore  choose  two  of  these  surfaces,  dis- 
tribute the  charge  of  A^  on  the  outer  of  these,  and  the  charge 
of  A.y  on  the  inner,  by  the  aid  of  the  principles  laid  down  in 
Section  50,  so  as  to  leave  the  values  of  the  potential  function 
on  these  surfaces  the  same  as  before.  These  distributions  thus 
found  will  remain  unchanged  if  the  equipotential  surfaces  are 
made  conducting. 

The  reader  who  wishes  to  study  this  method  more  at  length 
should  consult,  under  the  head  of  Electric  Images,  the  works  of 
Gumming,  INIaxwell,  Mascart,  and  Watson  and  Hurbury,  as  well 
as  original  papers  on  the  subject  by  ^Iuri)hy  in  the  Philosophical 
Magazine,  1833,  p.  350,  and  by  Sir  W.  Thomson  in  the  Cam- 
bridge and  Dublin  Mathematical  Journal  for  1848. 


134  ELECTEOSTATICS. 

67.  The  Energy  of  Charged  Conductors.  If  a  conductor  of 
capacity  C,  removed  from  the  action  of  all  electricity  except  its 
own,  be  charged  with  Mi  units  of  electricity,  so  that  it  is  at 

M 

potential  Vi  =  — -\  the  amount  of  work  required  to  bring  up  to 

the  conductor,  little  by  little,  from  the  walls  of  the  room,  the 
additional  charge  Am,  is  A  W,  which  is  greater  than  Vi  •  AJf  or 

^ .  AJf,  and  less  than  (  Vy  +  A^F)  •  AM  or  ¥l±A^.  AJf. 

If  the  charge  be  increased  from  Mi  to  M2  by  a  constant  flow, 
the  amount  of  work  required  is  evidently 

r^^MdM^M..^-Mi^^C 
Jm,       C  2  0  2  ^    '         '  ^  '-       -■ 

The  work  required  to  bring  up  the  charge  M  to  the  conductor 
at  first  uncharged  is  then 

M^^Cr^^MV^  [185] 

20  2  2  ■-       -■ 

This  is  evidently  equal  to  the  potential  energy  of  the  charged 
conductor,  and  this  is  independent  of  the  method  by  which  the 
conductor  has  been  charged. 

If,  now,  we  have  a  series  of  conductors  ^,,  Ao,  A^^  etc.,  in  the 
presence  of  each  other  at  potentials  V^  F2,  F,  etc.,  and  having 
respectively  the  charges  Mi,  Mo,  M^,  etc.,  and  if  we  change  all 
the  charges  in  the  ratio  of  ic  to  1,  we  shall  have  a  new  state  of 
equilibrium  in  which  the  charges  are  x3fi,  xMo,  xM^,  etc.  ;  and 
the  values  of  the  potential  functions  within  the  conductors  are 
xVi,  XV2,  .tFj,  etc.  The  work  (A  TF)  required  to  increase  the 
charges  in  the  ratio  x  +  Ax  instead  of  in  the  ratio  x  is  greater 
than 

{Ml  Ax)  {x  Vi)  -\-  (Mo  Ax)  (x  V2)  +  (3/2  Ax)  (x  Fi,)  +  etc. , 

or  X Ax[3Ii  Vi  +  M.2 V^  +  M^Vz  +  etc.] , 

and  less  than 

{x  +  Ax)AxlMiVi+M2V2-\-M^Vi-^Qtc.']  ; 


ELECTROSTATICS.  135 

hence  the  whole  amount  of  work  required  to  change  the  ratio 

from  —  to  -^  is 
1         1 

W, -  W,=  ?t^' [3/^ Yi  +M, V^  +M^Vs  +  etc.] .     [186] 

If  in  this  equation  we  put  Xj  =  0  and  a^,  =  1 »  we  get  for  the 
work  required  to  charge  the  conductor  from  the  neutral  state  to 
potentials  Fi,  F2,  F3, 

68.  If  a  series  of  conductors  Ai,  A21  A^,  etc.,  are  far  enough 
apart  not  to  be  exposed  to  inductive  action  from  one  another, 
and  have  capacities  Ci,  C2,  C3,  etc.,  and  charges  J/,,  J/,,  J/,,  etc., 
so  as  to  be  at  potentials  Fi,  Fq,  V,,  etc.,  where  3/i=C,Fi, 
M2  =  C2F2,  ^1/3=  C3F3,  etc.,  we  may  connect  them  together  by 
means  of  fine  wires  whose  capacities  we  may  neglect,  and  thus 
obtain  a  single  conductor  of  capacity 

The  charge  on  this  composite  conductor  is  evidently 

'     j/i  +  3/2  +  3/3  +  -  =^(^^n ; 

and  if  we  call  the  value  of  the  potential  function  within  it  F,  we 
shall  have  .^^  .^-^ 

F-2^(C)  =2^(30; 

whence  F=  -^>,^;r  T^Tr,    '  L  ^  ^^J 

Li  -h  O2  ■+•  L'.-i  ■+■  ••• 

a  formula  obtained,  it  is  to  be  noticed,  on  the  assumption  that 
the  conductors  do  not  influence  each  othor. 

The  energy  of  the  separate  charged  conductors  before  being 
connected  together  was 

TF=i(3/i  F:+3/2F2  +  J/3 1-3+ ••••)=  V^^^+^'  +  -^^+-) 

— Y  [189] 


=  ^I 


136  ELECTROSTATICS. 

and  the  energy  of  the  composite  conductor  is 


Ci+a  +  Ca-f  ... 


^[Zw. 


Z(^) 


[190] 


which  is  always  less  than  E  unless  the  separate  conductors  were 
all  at  the  same  potential  in  the  beginning. 


EXAMPLES. 

1.  Show  that  in  general  the  surface  density  of  a  charge  dis- 
tributed on  a  conductor  is  greatest  at  points  where  the  convex 
curvature  of  the  surface  of  the  conductor  is  greatest. 

2.  A  hollow  in  a  conductor  is  at  the  uniform  potential  V^ 
when  a  charge  is  communicated  to  a  conductor  within  the  cavity 
sufficient  to  raise  this  conductor  to  potential  Vo  if  it  were  in 
empty  space.  Give  some  idea  of  the  changes  brought  about  by 
this  charge. 

3.  Show  that  a  field  of  electric  force  consists  wholly  of 
non-conductors  bounded,  if  at  all,  by  conducting  surfaces. 

4.  Prove  that  if  a  distribution  of  electricity  over  a  closed 
surface  produce  a  force  at  every  point  of  the  surface  perpendic- 
ular to  it,  this  distribution  will  produce  a  potential  function  con- 
stant within  the  surface. 

5.  Two  conducting  spheres  of  radii  6  and  8  respectively 
are  connected  by  a  long  fine  wire,  and  are  supposed  not  to  be 
exposed  to  each  other's  influences.  If  a  charge  of  70  units  of 
electricity  be  given  to  the  composite  conductors,  show  that  30 
units  will  go  to  charge  the  smaller  sphere  and  40  units  to  the 
larger  sphere,  if  we  neglect  the  capacity  of  the  wire.     Show 

also  that  the  tension  in  the  case  of  the  smaller  sphere  is  -— - 

2oo7r 


per  square  unit  of  surface. 


ELECTROSTATICS.  137 

6.  An  uncharged  sphere  A,  of  radius  r,  occupies  the  centre 
of  the  otherwise  empty,  equipotential  cavit},  enclosed  by  a 
spherical  shell  B  of  radii  Vi  and  r„,  so  large  that  the  eflfect  inside 
the  cavity  of  the  charge  induced  on  B  by  a  charge  m,  communi- 
cated to  A  from  without,  may  be  neglected.  If  the  value  of  the 
potential  function  within  the  cavity  before  A  was  charged  was 

C,  at  what  potential  is  A  now?     Ans.   C  +  ^^—. 

r 

7.  The  first  of  three  conducting  spheres,  ^1,  B,  and  C,  of 
radii  3,  2,  and  1  respectively,  remote  from  one  another,  is 
charged  to  potential  9.  If  ^1  l)e  connected  with  B  for  an 
instant,  by  means  of  a  fine  wue,  and  if  then  B  be  connected 
with  C  in  the  same  way,  C's  charge  will  be  3-G.  [Stone.]  If, 
in  the  last  example,  all  three  conductors  be  connected  at  the 
same  time,  Cs  charge  will  be  4-."). 

8.  A  charge  of  M  units  of  electricity  is  communicated  to  a 
composite  conductor  made  up  of  two  widely-separated  ellipsoidal 
conductors,  of  semiaxes  2,  .3,  4  and  4,  G,  8  respectively,  con- 
nected by  a  fine  wire.  Show  that  the  charges  on  the  two  ellip- 
soids will  be   -3/ and  -  Jf  respectively.      [Stone.] 

;")  5 

9.  Can  two  electrified  bodies  repel  each  other  when  no  lines 
of  force  can  be  drawn  from  one  l)od\-  to  tlie  other? 

10.  Two  conductors,  ^l  and  7i,  connected  with  the  earth  are 
exposed  to  the  inductive  action  of  a  tiiird  charged  body.  Do 
A  and  B  act  upon  each  other?     If  so,  how? 

11.  Show  that  two  equal  conductors  similarh-  placed  with  re- 
spect to  each  other  always  rei)el  each  other  if  raised  to  the  same 
potentials  and  insulated  ;  Vuit  tiiat  if  the  volume  of  the  i)oten*^ial 
function  witliin  the  conductors  differ  never  so  little  from  each 
other,  they  will  repel  each  other  at  great  distances,  Imt  at  very 
near  distances  (supposing  no  spark  to  i)ass)  they  will  iittraet 
each  other.      [C'ummings.] 

12.  The  sui)erficial  density  has  the  same  sign  at  all  points  of 
a  conducting  surface  outside  which  tliere  is  no  free  electricity. 

13.  Show  that  r-^8  of  the  unit  tubes  of  force  proceeding 


138  ELECTROSTATICS. 

from  an  electrified  particle,  at  a  distance  8  from  the  centre  of  a 
conducting  spliere  of  radius  r,  which  is  put  to  earth,  meet  the 
sphere  if  tliere  are  no  other  conductors  in  the  neighborhood, 
and  that  tlie  rest  go  off  to  "infinity." 

14.  A  charged  insulated  conductor  A  is  so  surrounded  by  a 
number  of  sepai-ate  conductors  B,  C,  i),  •••,  wliich  are  put  to 
earth,  that  no  straight  line  can  be  drawn  from  any  point  of  A 
to  the  walls  of  the  room  without  encountering  one  of  these  other 
conductors :  will  there  be  any  induced  charge  on  the  walls  of 
the  room?     See  Section  37. 

15.  Two  uniform  straight  wires  of  equal  density,  each  two 
inches  long,  lie  separated  by  an  interval  of  one  inch  in  the 
same  straight  line.  Find  the  equation  of  the  equipotential  sur- 
faces due  to  these  wires,  and  find  what  must  be  the  density  of  a 
superficial  distribution  of  matter  on  one  of  tliese  surfaces  which 
at  all  outside  points  would  exert  the  same  attraction  as  the 
wires  do. 

16.  An  insulated  conducting  sphere  of  radius  r  charged  with 
m  units  of  positive  electricity  is  influenced  by  m  units  of  posi- 
tive electricity  concentrated  at  a  point  2r  distant  from  the  cen- 
tre of  the  sphere.  Give  approximately  the  general  shape  of  the 
equipotential  surfaces  in  the  neighborhood  of  the  sphere. 

Give  an  instance  of  a  positively  electrified  body  whose  poten- 
tial is  negative, 

17.  A  conductor,  the  equation  of  whose  surface  is 

^  +  ^+^1=1, 
25      16       9 

is  charged  with  80  units  of  electricity  ;  what  is  the  density  at  a 
point  for  which  a;  =  3,  y  —  S? 

If  tlie  density  at  this  point  be  a,  what  is  the  whole  charge  on 
tlie  ellipsoid? 

18.  Prove  that  the  capacity  of  n  equal  spherical  condensers 

when  arranged  in  cascade  is  onlv  about  -th  of  the  capacitv  of 

n 

one  of  the  condensers  ;  but  that  if  the  inner  spheres  of  all  the 


ELECTROSTATICS.  139 

condensers  be  connected  together  by  fine  wires,  and  the  outer 
conductors  be  also  connected  together,  the  capacity  of  the  com- 
plex condenser  thus  found  is  about  71  times  that  of  a  single 
one  of  the  condensers. 

19.  Prove  that  if  the  charges  of  a  system  of  conductors  be 
increased,  the  increment  of  the  energy  of  the  .system  is  equal  to 
half  the  sum  of  the  products  of  the  increase  in  the  charge  first 
conducted  into  the  sum  of  the  values  of  the  potential  function 
within  it  at  the  beginning  and  the  end  of  the  process,  or  to  half 
the  sum  of  the  products  of  tlie  increment  of  the  value  of  the 
potential  function  in  each  conductor  into  the  sum  of  the  original 
and  final  charges  on  that  conductor.      [Maxwell.] 

20.  Prove  that  if  the  charges  of  a  fixed  system  of  conductors 
be  increased,  the  sura  of  the  products  of  the  original  charge  and 
the  final  potential  of  each  conductor  is  equal  to  the  sum  of  the 
products  of  the  final  charge  and  the  original  potential.  [Max- 
well.] 

21.  Discuss  the  following  passage  from  Maxwell's  Elementarif 
Treatise  on  Electricity  : 

'•  Let  it  be  required  to  determine  the  equipotential  surfaces 
due  to  the  electrification  of  the  conductor  C  placed  on  an  insu- 
lating stand.  Connect  the  conductor  with  one  electrode  of  tlu' 
electroscope,  the  other  being  connected  with  tlie  earth.  Elec- 
trify the  exploring  sphere,*  and,  carrying  it  by  the  insulating 
handle,  bring  its  centre  to  a  given  point.  Connect  tlie  elec- 
trodes for  an  instant,  and  then  move  the  s|)here  in  such  a  path 
that  the  indication  of  the  electroscope  remains  zero.  This  path 
will  lie  on  an  equipotential  surface." 

22.  Pi'ove  tliat  the  coetlicients  of  potential  (/))  and  induction 
{(j)  treated  in  Article  /)i>  iiave  the  following  properties  : 

(1)  The  order  of  the  suffixes  of  any  p  or  any  7  can  be  invtMted 
without  altering  the  value  of  the  coetlicient.  or.  in  otlier  words. 


*  A  very  small  couducting  sphere  fitted  with  au  insulating  handle. 


140  ELECTROSTATICS. 

(2)  All  the  jp's  are  positive,  but  p,,.  is  less  than  either  p^  or  p^^. 

(3)  Those  c/s  whose  two  suffixes  are  the  same  are  positive  ; 
the  others  are  negative.  That  is,  g^j  and  Qu  are  positive  ;  but 
^y  is  negative  and  is,  moreover,  numerically  less  than  either  of 
the  others. 

23.  Prove  that  the  following  theorems  (Maxwell's  Elemen- 
tary Treatise  on  ElectHcity)  are  contained  in  the  statements  of 
the  preceding  problem  : 

(1)  In  a  system  of  fixed  insulated  conductors,  the  potential 
function  in  Aj,  due  to  a  charge  communicated  to  Ai  is  equal  to 
the  potential  function  in  Ai  due  to  an  equal  charge  in  A^. 

(2)  In  a  system  of  fixed  conductors  connected,  all  but  one, 
with  the  walls  of  the  room,  the  charge  induced  on  A^  when  Ai 
is  raised  to  a  given  potential  is  equal  to  the  charge  induced  on 
Ai  when  A^  is  raised  to  an  equal  potential. 

(3)  If  in  a  S3"stem  of  fixed  conductors,  insulated  and  origi- 
nalh'  without  charge,  a  charge  be  communicated  to  A^  which 
raises  it  to  potential  unity  and  Ai  to  potential  n,  then  if  in  the 
same  system  of  conductors  a  charge  unity  be  communicated 
to  Ai,  and  A^  be  connected  with  the  earth,  the  charge  induced 
on  A,,  will  be  —  w. 

24.  A  condenser  consists  of  a  sphere  A  of  radius  100  sur- 
rounded by  a  concentric  shell  whose  inner  radius  is  101  and 
outer  radius  150.  The  shell  is  put  to  earth,  and  the  sphere  has 
a  charge  of  200  units  of  positive  electricity.  A  sphere  B  of 
radius  100  outside  the  condenser  can  be  connected  with  the 
condenser's  sphere  by  means  of  a  fine  insulated  wire  passing 
through  a  small  hole  in  the  shell.  B  is  connected  with  A ;  tlie 
connection  is  then  broken,  and  B  is  discharged  ;  the  connection 
is  then  made  and  broken  as  before,  and  B  is  again  discharged. 
After  this  process  has  been  gone  through  with  five  times,  what  is 
yl's  potential?  What  would  it  become  if  the  shell  were  to  be 
removed  without  touching  A'i 

25.  Suppose  the  condenser  mentioned  in  the  last  problem  in- 
sulated and  a  charge  of  100  units  of  positive  electricity  given  to 


ELECTllOSTATICS.  141 

the  shell.  What  will  be  the  potential  of  the  sphere?  of  the 
shell  ?  If  we  then  connect  the  sphere  with  the  earth  b}'  a  fine 
insulated  wire  passing  through  the  shell,  what  will  the  charge  on 
tlie  shell  be?  "What  will  be  the  potential  of  tlie  shell?  If  next 
A  be  insulated,  and  the  shell  be  put  t«i  earth,  what  will  be  A's 
potential  ?  What  will  be  its  potential  if  the  shell  ])e  now  wholly 
removed  ? 

26.  A  spherical  conductor  of  radius  r  is  surrounded  by  a  con- 
centric conducting  spherical  shell  of  radii  7?,  and  7?,,,  and  the 
outer  surface  of  this  shell  is  put  to  earth.  If  the  inner  conduc- 
tor be  charged,  show  the  effect  at  all  points  in  space  of  moving 
the  conductor  so  that  it  shall  be  eccentric  with  the  shell.  How 
is  the  capacity  of  the  system  changed  by  this  ? 

27.  Prove  that  if  the  spherical  surfaces  of  radii  a  and  b, 
which  form  a  spherical  condenser,  are  made  slightly  eccentric, 
c  being  the  distance  between  their  centres,  the  change  of  elec- 
trification at  anv  point  of  either  surface  is  '- — '- , 

'   ^  47r(6-a)(6^-a^) 

where  0  is  the  angular  distance  of  the  point  from  the  line  of 
centres,  and  where  the  difference  between  the  values  of  the 
potential  function  on  the  two  surfaces  is  unity. 

28.  Show  that  if  an  insulated  conducting  sphere  of  radius  a 
be  placed  in  a  region  of  uniform  force  (X),  acting  parallel  to 


the  axis  of  x,  the  function  —  X 


1-- 


+  C  satisfies  all  the 


conditions  which  the  potential  function  outside  the  sphere  must 

satisfy,  and   is  therefore  itself  the  potential  function.     Show 

SxX 
that  the  surface  density  of  the  charge  on  the  sphere  is  — —. 

[Watson  and  Burbury.] 


142  ELECTROSTATICS. 


MISCELLANEOUS    PROBLEMS. 


1.  Prove  that  the  attraction  due  to  a  homogeneous  hemi- 
Bphere  of  radius  r  is  zero,  at  a  point  in  the  axis  of  the  hemisphere 

3 
distant  -r  approximately  from  the  centre  of  the  base. 

2.  Show  that  the  attraction  at  the  origin  due  to  the  homo- 
geneous sohd  bounded  by  the  surface  obtained  b^'  revolving 
one  loop  of  the  curve  r^=.o?  •  cos  2^,  is  \  -n-a. 

3.  If  the  earth  be  considered  as  a  homogeneous  sphere  of 
radius  r,  and  if  the  force  of  gravity  at  its  surface  be  gr,  show 
that  from  a  point  without  the  earth,  at  which  the  attraction  is 

99  —  1  I  1}         I   1  \ 

gf,  the  area  27rr^(  1  —  • )  on  the  surface  of  the  earth 

n  \  *^    / 

will  be  visible. 

4.  A  spherical  conductor  A,  of  radius  ci,  charged  with  M 
units  of  electricity,  is  surrounded  by  n  conducting  spherical 
shells  concentric  with  it.  Each  shell  is  of  thickness  a,  and  is 
separated  from  its  neighbors  by  empty  spaces  of  thickness  a. 
Show  that  the  limit  approached  b}'  V^  as  n  is  made  larger  and 

M 

larger  is  —  (nat.  log  2) ,  and  that  for  a  finite  number  of  shells 
a 

K^  =  —  I    — ■ ax.     [Stone.] 

a  Jo       l+x 

5.  If  two  systems  of  matter  {M  and  3f' ),  both  shut  in  by  a 
closed  surface  S^  give  rise  to  potential  functions  ( V  and  V) , 
which  have  equal  values  at  every  point  of  S,  whether  or  not  8 
is  an  equipotential  surface  of  either  system,  then  V  cannot 
differ  from  V  at  any  point  outside  S,  and  the  algebraic  sum  of 
the  matter  of  either  system  is  equal  to  that  of  the  other.  [See 
Section  52,  and  Watson  and  Burbury's  Mathematical  Theory  of 
Electricity  and  Magnetism.,  §  60.] 

6.  Show  that  if  two  distributions  of  matter  have  in  common 
an  equipotential  surface  which  surrounds  them  both,  all  their 
equipotential  surfaces  outside  this  will  be  common. 


ELECTKOSTATICS.  143 

7,  Prove  that  if  V  be  the  potential  function  due  to  any  dis- 
tribution of  matter  over  a  closed  surface  >S',  and  if  a-'  be  tiie 
density  of  a  superficial  distribution  on  /S,  whicli  gives  r.se  to 
the  same  value  of  the  potential  function  at  eacli  point  of  S  as 
that  of  a  unit  of  matter  concentrated  at  any  given  point  0, 
tlien  the  value  at  0  of  the  potential  function  due  to  the  first 

distribution  is  |  V-a-'  •  dS. 


^rtss  of 
$oston. 


158 

PeircR's  Three  and  Four  Place  Tables  of  Loga- 

rithiiiic  and  '/'ri^oitomftric  Functions.  By  Jami-.s  Mri.i.s  ]'i:ii;cK, 
University  I'rofessur  of  Mathematics  in  Harvard  University.  Quarto. 
Cloth.     Mailing  Trice,  45  cts. ;   Introduction,  40  cts. 

Four-place  tables  require,  in  the  long  run,  only  half  a.s  mucli  lime 
#  five-place  tables,  one-thiid  as  much  time  as  si.x-place  tables,  and 
ane-fourth  as  much  as  those  of  seven  places.  They  are  sufficient 
for  the  ordinary  calculations  of  Surveying,  Civil,  Mechanical,  and 
Mining  Engineering,  and  Navigation;  for  the  work  of  the  Physical 
or  Chemical  Laboratory,  and  even  for  many  computations  of  Astron- 
omy. They  are  also  especially  suited  to  be  used  in  teaching,  as  they 
illustrate  principles  as  well  as  the  larger  tables,  and  with  far  less 
expenditure  of  time.  The  present  compilation  has  been  prepared 
with  care,  and  is  handsomely  and  clearly  printed. 

Elements  of  the  Differential  Calculus. 

With  Numerous  Examples  and  Aj^plications.  De>ij;ned  for  Use  as  a 
College  Text-Book.  I3y  \V.  E.  Bveki.v,  I'rofessor  of  Mathematics, 
Harvard  University.  8vo.  273  pages.  .Mailing  Price,  $2.15  ;  Intro- 
duction, $2.00. 

This  book  embodies  the  results  of  the  author's  experience  in 
teaching  the  Calculus  at  Cornell  and  Harvard  Universities,  and  is 
intended  for  a  text-book,  and  not  for  an  exhaustive  treatise.  Its 
peculiarities  are  the  rigorous  use  of  the  Doctrine  of  Limits,  as  a 
foundation  of  the  .subject,  and  as  preliminary  to  the  adojnion  of  the 
more  direct  and  practically  convenient  infinitesimal  notation  and 
nomenclature;  the  early  introduction  of  a  few  simple  formulas  and 
methods  for  integrating:  a  rather  elaborate  treatment  of  the  use  of 
infinitesimals  in  pure  geometry ;  and  the  attempt  to  excite  and  keep 
up  the  interest  of  the  stutient  by  bringing  in  throughout  the  whole 
book,  and  not  merely  at  the  end,  numerous  applicitions  to  practical 
problems  in  geometry  and  mechanics. 

James  Mills  Peirce,  Pmf.  c/lis  general  without  being  superficial; 
Math.,  HtivarJ  Ihnv.  (From  tlie  Har-\  linuied  to  le.idinij  topics,  and  yet  with- 
vard  R'e::isfi-r)  :  In  mathematics,  as  in  in  its  limits;  tliorough.  accurate,  and 
other  tir.inches  of  study,  the  need  is  practical;  adapted  to  the  communica- 
now  very  much  felt  of  teaching  which    tion  of  some  degree  of  power,  as  well 


159 


as  knowledge,  but  free  from  details 
which  are  important  only  to  the  spe- 
cialist. Professor  Byerly's  Calculus 
appears  to  be  designed  to  meet  this 
want.  .  .  .  Such  a  plan  leaves  much 
room  for  the  exercise  of  individual 
judgment ;  and  differences  of  opinion 
will  undoubtedly  exist  in  regard  to  one 
and  another  point  of  this  book.  But 
all  teachers  will  agree  that  in  selection, 
arrangement,  and  treatment,  it  is,  on 
the  whole,  in  a  very  high  degree,  wise, 
able,  marked  by  a  true  scientific  spirit, 
and  calculated  to  develop  the  same 
spirit  in  the  learner.  .  .  .  The  book 
contains,  perhaps,  all  of  the  integral 
calculus,  as  well  as  of  the  differential, 
that  is  necessary  to  the  ordinary  stu- 
dent. And  with  so  much  of  this  great 
scientific  method,  every  thorough  stu- 
dent of  physics,  and  every  general 
scholar  who  feels  any  interest  in  the 
relations  of  abstract  thought,  and  is 
capable  of  grasping  a  mathematical 
idea,  ought  to  be  familiar.  One  who 
aspires  to  technical  learning  must  sup- 
plement his  mastery  of  the  elements 
by  the  study  of  the  comprehensive 
theoretical  treatises.  .  .  .  But  he  who  is 
thoroughly  acquainted  with  the  book 
before  us  has  made  a  long  stride  into 
a  sound  and  practical  knowledge  of 
the  subject  of  the  calculus.  He  has 
begun  to  be  a  real  analyst. 

H.  A.  Newton,  Prof,  of  Math,  in 
Vale  Coll.,  A'i-io  Haven  :  1  have  looked 
it  through  with  care,  and  find  the  sub- 
ject very  clearly  and  logically  devel- 
oped. 1  am  strongly  inclined  to  use  it 
in  my  class  next  year. 

S.  Hart,  receni  Prof,  of  Math,  in 
Trinity  Coll.,  Conn.:  The  student- can 
hardly  fail,  I  think,  to  get  from  the  book 
an  exact,  and,  ;it  the  same  time,  a  satis- 
factory expian.ition  of  the  principles  on 
which  the  Calculus  is  based ;  and  the 
introduction  of  the  simpler  methods  of 


integration,  as  they  are  needed,  enables 
applications  of  tliose  principles  to  be 
introduced  in  such  a  way  as  to  be  both 
interesting  and  instructive. 

Charles  Kraus,  Techniker,  Pard- 
iibitz,  Bohemia,  Austria  :  Indem  ich 
den  Empfang  Ihres  Buches  dankend 
bestaetige  muss  ich  Ihnen,  hoch  geehr- 
ter  Herr  gestehen,  dass  mich  dasselbe 
sehr  erfrcut  hat,  da  es  sich  durch 
grosse  Reichhaliigkeit.besonders  klare 
Schreibweisc  und  vorzuegliche  Behand- 
lung  des  Stoffes  auszeichnet,  und  er- 
weist  sich  dieses  Werk  als  eine  bedeut- 
ende  Bereicherung  dermathematischen 
Wissenschaft. 

De  Volson  Wood,  Prof,  of 
Math.,  Stevens'  Inst.,  Hoboken,  N.J. : 
To  say,  as  1  do,  that  it  is  a  first-class 
work,  is  probably  repeating  what  many 
have  already  said  for  it.  I  admire  the 
rigid  logical  character  of  the  work, 
and  am  gratified  to  see  that  so  able  a 
writer  has  shown  explicitly  the  relation 
between  Derivatives,  Infinitesimals,  and 
Differentials,  The  method  of  Limits 
is  the  true  one  on  which  to  found  the 
science  of  the  calculus.  The  work  is 
not  only  comprehensive,  but  no  vague- 
ness is  allowed  in  regard  to  definitions 
or  fundamental  principles. 

Del  Kemper,  Prof  of  Math., 
Hampden  Sidney  Coll..  I 'a.:  My  high 
estimate  of  it  has  boen  amply  vindi- 
cated by  its  use  in  the  class-room. 

R.  H.  Graves,  Prof  of  Math., 
Univ.  of  J^'orth  Carolina  :  I  have  al- 
ready decided  to  use  it  with  my  next 
class ;  it  suits  my  purpose  better  than 
any  other  book  Cn  the  same  subject 
with  which  I  am  acquainted. 

Edw.  Brooks,  Author  of  a  Series 
of  Math. :  Its  statements  are  clear  and 
scholarly,  and  its  methods  thoroughly 
analytic  and  in  the  spirit  of  the  latest 
mathematical  thought. 


i6o 


Syllabus  of  a  Course  in  Plane  Trigonometry. 

By  \V.  K.  1!vi:ki.v.     8vo.     8  pages.     Mailing  Price,  lo  cts. 

Syllabus  of  a  Course  in  Plane  Analytical  Geom- 

etry.     JJy  \V.  E.  Bverly.     Svo.      12  pages.     Mailing  Price,  lo  cts. 

Syllabus  of  a  Course  in  Plane  Analytic  Geom- 

etry     {Advanced  Course.)     By   W.    E.   Byerly,   Professor  of  Mathe- 
matics, Harvard  University.     Svo.      12  pages.     Mailing  Price,  10  cts. 

Syllabus  of  a  Course  in  Analytical  Geometry  of 

Three  Dimensions.     By   W.   E.   BVEKI.Y.     Svo.      10  pages.      Mailing 
Price,  10  cts. 

Syllabus   of  a   Course  on   Modern  Methods   in 

Analytic  Geometry.     By   W.   E.  Byerly.      Svo.      S  pages.      Mailing 
Price,  10  cts. 

Syllabus  of  a  Course  in  the  Theory  of  Equations. 

By  \V.  E.  Byeri.y.     Svo.     S  pages.     Mailing  Price,  10  cts. 

Elements  of  the  Integral  Calculus. 

By  W.  E.  Byeri.y,  Professor  of  Matliematics  in  Ilarv-ard  University. 
Svo.     204  pages.     Mailing  Price,  $2. 15;   Introduction,  S2.00. 

This  volume  is  a  sequel  to  the  author's  treatise  on  the  DitTercntial 
Calculus  (see  page  134),  and,  like  that,  is  written  as  a  tc.xt-book. 
The  last  chapter,  however,  —  a  Key  to  the  Solution  of  Differential 
Equations,  —  may  prove  of  service  to  working  mathematicians. 


H.  A.  Newton,  Prof,  of  Math., 
Yale  Coll.:  We  shall  use  it  in  my 
optional  class  next  term. 

Mathematical    Visitor :     The 

subject  is  presented  very  clearly.  It  is 
liie  first  American  treatise  on  the  Cal- 
culus that  we  have  seen  which  devotes 
any  space  to  average  and  probability. 

Schoolmaster,  London :  The 
merits  of  this  work  are  as  marked  as 


those  of  the  DifTercntial   Calculus  by 
the  same  author. 

Zion's  Herald  :  .\  text-book  every 
way  worthy  of  the  venerable  University 
in  which  the  author  is  an  honored 
teacher.  C.imbriiige  in  Massachusetts, 
like  Cambridge  in  England,  preserves 
its  reputation  for  the  breadth  and  strict- 
ness of  its  malheinaticai  requisitions, 
and  these  form  the  spinal  column  of  a 
liberal  education. 


i6i 


>4  Short  Table  of  Integrals. 


To  accompany  BYERLY'S  INTEGRAL  CALCULUS.  By  B.  O. 
Peirce,  Jr.,  Instructor  in  Mathematics,  Harvard  University.  i6  pages. 
Mailing  Price,  lo  cts.   To  be  bound  with  future  editions  of  the  Calculus. 


Elements  of  Quaternions. 


By  A.  S.  Hardy,  Ph.D.,  Professor  of  Mathematics,  Dartmouth  College. 
Crown,  8vo.  Cloth.  240  pages.  Maihng  Price,  ^2.15;  Introduction, 
$2.00. 

The  chief  aim  has  been  to  meet  the  wants  of  beginners  in  the 
class-room.  The  Elements  and  Lectures  of  Sir  W.  R.  Hamilton 
are  mines  of  wealth,  and  may  be  said  to  contain  the  suggestion 
of  all  that  will  be  done  in  the  way  of  Quaternion  research  and 
application :  for  this  reason,  as  also  on  account  of  their  diffuseness 
of  style,  they  are  not  suitable  for  the  purposes  of  elementary  instruc- 
tion. The  same  may  be  said  of  Tait's  Qiiaterniotts,  a  work  of 
great  originality  and  comprehensiveness,  in  style  very  elegant  but 
very  concise,  and  so  beyond  the  time  and  needs  of  the  beginner. 
The  Introduction  to  Quaternions  by  Kelland  contains  many  exer- 
cises and  examples,  of  which  free  use  has  been  made,  admirably 
illustrating  the  Quaternion  spirit  and  method,  but  has  been  found, 
in  the  class-room,  practically  deficient  in  the  explanation  of  the 
theory  and  conceptions  which  underlie  these  applications.  Tlie 
object  in  view  has  thus  been  to  cover  the  introductory  ground  :iiore 
thoroughly,  especially  in  symbolic  transformations,  and  at  the  same 
time  to  obtain  an  arrangement  better  adapted  to  the  methods  of 
instruction  common  in  this  country. 

PRESS    NOTICES. 


Westminster  Review  :  It  is  a 

remarkably  clear  exposition  of  the  sub- 
,  ject. 

The  Daily  Review,  Edinburgh, 
Scotland :  This  is  an  admirable  text- 
book. Prof.  Hardy  has  ably  supplied 
a  felt  want.  The  definitions  are  models 
of  conciseness  and  perspicuity. 


The  Nation  :  For  those  who  have 
never  studied  the  subject,  this  treatise 
seems  to  us  superior  both  to  the  work 
of  Prof.  Tait  and  to  the  joint  treatise  by 
Profs.  Tait  and  Kelland. 

New  York  Tribune :  The  Qua- 
ternion Calculus  Is  an  instrument  of 
mathematical  research  at  once  so  pow- 


l62 


erful,  flexible,  and  elegant,  so  sweeping 
in  its  range,  and  so  minutely  accurate, 
that  its  discovery  and  development  has 
been  rightly  estimated  as  one  of  the 
crowning  achievements  of  the  century. 
The  time  is  approaching  when  all  col- 
leges will  insist  upon  its  study  as  an 
essential  part  of  the  equipment  of  young 
men  who  aspire  to  be  classified  among 
the  liberally  educated.  This  book  fur- 
nishes just  the  elementary  instruction 
on  the  subject  which  is  needed. 

New  York  Times :  It  is  especially 
designed  to  meet  the  needs  of  begin- 
ners in  the  science.  ...  It  has  a  way 
of  putting  things  which  is  eminently  its 
own,  and  which,  for  clearness  and  force, 
is  as  yet  unsurpassed.  ...  If  we  may 
not  seek  for  Quaternions  made  easy,  we 
certainly  need  search  no  longer  for 
Quaternions  made  plain. 

Van  Nostrand  Engineering 
Magazine :  To  any  one  who  has 
labored  with  the  very  few  works  ex- 
tant upon  this  branch  of  mathematics, 
a  glance  at  the  opening  chapter  of 
Prof.  Hardy's  work  will  enforce  the 
conviction  that  the  author  is  an  in- 
structor of  the  first  order.  The  book 
is  quite  opportune.    The  subject  must 


soon  become  a  necessary  one  in  all 
the  higher  institutions,  for  already  are 
writers  of  mathematical  essays  making 
free  use  of  Quaternions  without  any 
preliminary  apology. 

Canada   School  Jotomal,    7b- 

ronto :  The  author  of  this  treatise  has 
shown  a  thorough  mastery  of  the  Qua- 
ternion Calculus. 

London  Schoolmaster :  It  is  in 
every  way  suited  to  a  student  who 
wishes  to  commence  the  subject  ab 
initio.  One  will  require  but  a  few 
hours  with  this  book  to  learn  that  this 
Calculus,  with  its  concise  notation,  is  a 
most  powerful  instrument  for  mathe- 
matical operations. 

Boston  Transcript :  A  text-book 
of  unquestioned  excellence,  and  one 
peculiarly  fitted  for  use  in  American 
schools  and  colleges. 

The  Western,  St.  Louis:  This 
work  exhibits  the  scope  and  power  of 
the  new  analysis  in  a  very  clear  and 
concise  form  .  .  .  illustrates  very  finely 
the  important  fact  that  a  few  simple 
principles  underlie  the  whole  body  of 
mathematical  truth. 


FKOM    COLLEGE    PROFESSORS. 


James  Mills  Peirce,  Prof,  of 
Math.,  Harvard  Coll. :  I  am  much 
pleased  with  it.  It  seems  to  me  to 
supply  in  a  very  satisfactory  manner 
the  need  which  has  long  existed  of  a 
clear,  concise,  well-arranged,  and  logi- 
cally-developed introduction  to  this 
branch  of  Mathematics.  I  think  Prof. 
Hardy  has  shown  excellent  judgment 
in  his  methods  of  treatment,  and  also 
in  limiting  himself  to  the  exposition 
and  illustration  of  the  fundamental 
principles  of  his  subject.     It  is,  as  it 


ought  to  be,  simply  a  preparation  for 
the  study  of  the  writings  of  Hamilton 
and  Tait.  I  hope  the  publication  o.' 
this  attractive  treatise  will  increase  the 
attention  paid  in  our  colleges  to  the 
profound,  powerful,  and  fascinating  cal- 
culus of  which  it  treats. 

Charles  A.  Young,  Prof,  of 
Astronomy,  Princeton  Coll.  :  I  find  it 
by  far  the  most  clear  and  intelligible 
statement  of  the  matter  I  have  yet 
seen. 


t63 


Elements  of  the  Differential  and  Integral  Calculus. 

With  Examples  and  Applications.  By  J.  M.  Taylor,  Professor  of 
Mathematics  in  Madison  University.  8vo.  Cloth.  249  pp.  Mailinj; 
price,  $1.95;   Introduction  price,  $1.80. 

The  aim  of  .this  treatise  is  to  present  simply  and  concisely  the 
fundamental  problems  of  the  Calculus,  their  solution,  and  more 
common  applications.  Its  axiomatic  datum  is  that  the  change  of  a 
variable,  when  not  uniform,  may  be  conceived  as  becoming  uniform 
at  any  value  of  the  variable. 

It  employs  the  conception  of  rates,  which  affords  finite  differen- 
tials, and  also  the  simplest  and  most  natural  view  of  the  problem  of 
the  Differential  Calculus.  This  problem  of  finding  the  relative 
rates  of  change  of  related  variables  is  afterwards  reduced  to  that  of 
finding  the  fimit  of  the  ratio  of  their  simultaneous  increments ;  and, 
in  a  final  chapter,  the  latter  problem  is  solved  by  the  principles  of 
infinitesimals. 

Many  theorems  are  proved  both  by  the  method  of  rates  and  that 
of  limits,  and  thus  each  is  made  to  throw  light  upon  the  other. 
The  chapter  on  differentiation  is  followed  by  one  on  direct  integra- 
tion and  its  more  important  applications.  Throughout  the  work 
there  are  numerous  practical  problems  in  Geometry  and  Mechanics, 
which  serve  to  exhibit  the  power  and  use  of  the  science,  and  to 
excite  and  keep  alive  the  interest  of  the  student. 

Judging  from  the  author's  experience  in  teaching  the  subject,  it 
is  believed  that  this  elementary  treatise  so  sets  forth  and  illustrates 
the  highly  practical  nature  of  the  Calculus,  as  to  awaken  a  lively 
interest  in  many  readers  to  whom  a  more  abstract  method  of  treat- 
ment would  be  distasteful. 


Oren  Root,  Jr.,  Piof.  of  Math., 
Hamiltitn  Coll.,  N.Y.:  In  reading  the 
manuscript  I  was  impressed  by  the 
clearness  of  definition  and  demonstra- 
tion, the  pertinence  of  illusiration,  and 
the  liajipy  union  of  exchision  and  con- 
densation. It  seems  to  me  most  admir- 
ably suited  for  use  in  college  classes. 
I  jjrove  my  regard  by  adopting  this  as 
our  text-book  on  the  calculus. 


C.    M.    Charrappin,    S.J.,   SI. 

Louis  Univ. :  I  have  given  the  book  a 
thorough  examination,  and  I  am  satis- 
fied that  it  is  the  best  work  on  the  sub- 
ject I  have  seen.  I  mean  the  best 
work  for  what  it  was  intended, —  a  text- 
book. I  would  like  very  much  to  in- 
troduce it  in  the  University, 
{Jan.  12,  1885.) 


University  of  California 
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